Tuesday, February 5

Engineering 101

The year-ago ice jam hit the Raymore Park construction site hard. The area was the west-side-of-the-river staging point for an under-the-river pipe-refitting project.


A few months later things seemed back to normal. There were a couple of dirt mounds adjacent to the manhole entrance.


In light of the recurring ice jams on the river, an engineering student might have thought to distribute the dirt in situ so as to level the top of the manhole with the surrounding ground. But there were no such engineering students on hand, so they decided to haul the dirt away.


In late-September, early-October, a thin layer of soil — likely with embedded grass seed — was added to the site.


Here is how the manhole entrance appeared on October 16:


Here is how it appeared this morning after the current warm-spell ice melt:


I'm wondering if the top concrete layer has been sheared off.

February 8 update:

Monday, January 28

Pseudo-binary ænlic primes

I discussed ænlic primes here. By pseudo-binary I mean decimal numbers that contain only digits zero and one. I will cover here three types of pseudo-binary ænlic primes:

1. Repunit primes. These are terms of A004022. They are ænlic primes by default — no other permutation of their digits is prime (because no other permutation of their digits exists). The repunit number (10^a-1)/9 is an ænlic prime for a = 2, 19, 23, 317, 1031, 49081, 86453, 109297, 270343, ...

2. Primes composed of all ones except for one zero. The number (10^a-1)/9-10^b is an ænlic prime for (a,b) = (5,3), (6,4), (9,7), (26,13), (29,5), (33,13), (35,7), (39,23), (51,43), (53,37), (56,8), (74,44), (77,33), (80,11), (83,31), (89,39), (90,83), (92,53), (105,67), (107,11), (108,99), (110,61), (113,73), (126,44), (129,109), (134,80), (147,19), (149,15), (153,25), (164,15), (170,6), (174,62), (176,113), (177,167), (182,17), (185,37), (192,80), (194,26), (195,151), (201,53), (204,131), (215,119), (219,167), (221,181), (225,211), (227,223), (228,215), (230,32), (231,127), (234,208), (236,207), (246,76), (249,23), (251,147), (260,235), (263,151), (267,43), (270,128), (284,199), (285,53), (288,54), (297,13), (305,269), (314,278), (315,227), (317,125), (321,101), (324,320), (329,135), (330,289), (348,3), (363,83), (369,125), (371,131), (381,127), (387,379), (389,95), (396,262), (417,287), (419,119), (443,55), (446,342), (450,209), (453,421), (461,289), (467,459), (473,213), (477,53), (480,242), (491,31), (492,124), (497,423), (503,439), (504,486), (506,446), (512,83), (518,131), (521,419), (525,289), (539,483), (540,8), (545,247), (546,128), (548,284), (566,361), (567,103), (572,492), (579,83), (582,160), (588,396), (593,135), (605,79), (606,521), (609,163), (611,363), (618,208), (629,89), (632,293), (635,311), (641,331), (650,123), (653,33), (656,363), (657,175), (659,535), (674,626), (677,655), (681,509), (692,132), (698,209), (708,347), (710,25), (711,623), (713,451), (720,120), (729,697), (750,146), (752,98), (765,167), (770,530), (779,355), (780,84), (791,623), (803,459), (804,148), (806,13), (809,105), (816,336), (828,816), (833,491), (836,399), (837,425), (845,35), (848,521), (849,691), (857,545), (858,627), (861,293), (870,112), (872,475), (875,771), (885,703), (890,87), (893,85), (897,487), (912,35), (914,168), (923,71), (935,415), (939,295), (945,169), (954,897), (956,746), (969,311), (984,695), (993,397), (1001,259), (1017,151), (1019,735), (1020,843), (1022,629), (1029,923), (1037,289), (1040,11), (1052,559), (1067,863), (1070,750), (1077,695), (1083,631), (1085,463), (1086,1073), (1091,511), (1095,655), (1097,565), (1101,997), (1110,626), (1112,889), (1124,561), (1127,711), (1136,737), (1139,999), (1140,310), (1146,526), (1148,119), (1149,253), (1155,455), (1160,746), (1161,61), (1163,227), (1173,187), (1175,983), (1184,347), (1196,557), (1199,623), (1202,194), (1203,583), (1208,189), (1217,35), (1230,1190), (1232,277), (1235,959), (1254,1035), (1256,1083), (1259,1143), (1265,735), (1274,1199), (1277,1113), (1287,607), (1289,755), (1302,446), (1307,1235), (1320,239), (1328,853), (1331,139), (1338,905), (1346,1237), (1353,907), (1361,1329), (1365,635), (1373,1269), (1376,608), (1377,83), (1382,547), (1385,545), (1392,638), (1407,383), (1409,245), (1421,377), (1433,1013), (1451,1239), (1466,999), (1470,1159), (1472,1344), (1473,271), (1481,1059), (1485,325), (1487,527), (1509,467), (1511,1231), (1514,1039), (1515,443), (1517,1239), (1521,229), (1523,947), (1526,529), (1529,421), (1532,1205), (1533,749), (1541,1123), (1548,1388), (1550,828), (1553,1351), (1554,684), (1556,746), (1566,738), (1568,1001), (1572,1244), (1586,843), (1589,1017), (1598,168), (1604,289), (1613,319), (1617,139), (1620,4), (1625,69), (1635,1595), (1641,841), (1643,427), (1658,1299), (1662,1486), (1665,479), (1670,1214), (1686,191), (1688,114), (1689,637), (1700,524), (1706,3), (1707,383), (1709,1607), (1716,83), (1721,369), (1725,223), (1731,119), (1737,941), (1739,1435), (1757,1491), (1763,1019), (1766,893), (1770,104), (1773,661), (1793,601), (1796,1003), (1797,421), (1802,380), (1805,229), (1809,13), (1811,23), (1815,719), (1818,1181), (1821,751), (1836,177), (1850,143), (1862,1407), (1880,294), (1904,1195), (1905,1891), (1907,1839), (1914,1012), (1926,1226), (1929,473), (1934,1017), (1955,303), (1956,416), (1958,1081), (1970,1755), (1973,331), (1985,1355), (1986,1426), (1992,1083), (1995,535), (2010,1518), (2016,396), (2019,683), (2021,1977), (2034,1186), (2036,476), (2037,1451), (2039,1679), (2040,1270), (2048,543), (2058,2022), (2060,752), (2063,1071), (2067,1895), (2069,2045), (2070,2001), (2084,1065), (2085,1865), (2091,491), (2094,25), (2097,1463), (2105,1011), (2112,41), (2115,1007), (2123,531), (2124,1345), (2127,2063), (2135,79), (2136,1214), (2138,62), (2141,57), (2147,1515), (2166,1426), (2178,1320), (2192,120), (2193,509), (2195,1723), (2198,1739), (2204,895), (2205,611), (2208,1648), (2210,1697), (2213,1523), (2217,593), (2219,1643), (2225,481), (2234,755), (2235,991), (2249,2123), (2255,1711), (2259,467), (2261,1909), (2262,13), (2264,2113), (2265,1351), (2289,823), (2306,1038), (2309,1305), (2316,594), (2321,553), (2324,1089), (2325,907), (2328,1015), (2336,1403), (2339,1703), (2345,665), (2354,901), (2366,1886), (2373,461), (2379,107), (2385,1225), (2387,159), (2388,1713), (2390,1353), (2394,152), (2397,1295), (2400,463), (2403,203), (2409,5), (2411,459), (2417,257), (2420,1700), (2426,1389), (2430,2106), (2432,857), (2433,751), (2439,2239), (2441,599), (2447,943), (2451,1543), (2457,1609), (2465,1339), (2472,1648), (2478,232), (2481,1781), (2486,1817), (2489,969), (2490,1304), (2498,1541), (2501,2117), (2508,864), (2513,673), (2519,1679), (2520,223), (2522,2466), (2526,1522), (2528,2067), (2535,1567), (2537,1889), (2546,108), (2549,757), (2550,1175), (2552,889), (2553,1933), (2561,863), (2564,2363), (2565,359), (2567,2039), (2571,839), (2574,1322), (2577,1129), (2585,967), (2594,313), (2595,607), (2597,2019), (2598,974), (2601,2329), (2606,1229), (2610,1192), (2612,1987), (2616,1281), (2618,1827), (2621,2607), (2628,1762), (2633,2073), (2636,903), (2646,1011), (2661,1321), (2670,810), (2673,2095), (2688,2672), (2693,1033), (2694,1187), (2696,2568), (2708,855), (2733,1537), (2759,1119), (2766,2361), (2769,383), (2774,558), (2784,1545), (2789,1019), (2793,2023), (2795,1863), (2796,2563), (2802,2724), (2804,1610), (2807,1455), (2816,449), (2822,356), (2826,1362), (2828,635), (2844,2126), (2846,2597), (2856,2573), (2858,593), (2861,1197), (2865,991), (2867,991), (2873,1801), (2877,1085), (2882,1454), (2883,779), (2885,1065), (2889,1579), (2897,2701), (2903,1031), (2904,894), (2912,165), (2925,197), (2927,447), (2931,1063), (2939,2755), (2948,727), (2954,2369), (2963,2711), (2966,1316), (2981,421), (2987,1155), (2993,555), (2996,608), (2997,2777), (3008,1807), (3009,2405), (3020,2891), (3027,823), (3033,361), (3041,1847), (3044,2694), (3047,1359), (3066,2488), (3075,3031), (3080,1537), (3081,817), (3083,2107), (3111,2519), (3114,12), (3116,848), (3126,1936), (3128,1927), (3129,343), (3131,2847), (3135,1663), (3137,2399), (3156,2089), (3165,2035), (3170,896), (3173,875), (3186,2354), (3191,1047), (3201,1219), (3209,2407), (3213,2951), (3233,2885), (3239,983), (3242,1188), (3246,1819), (3249,3173), (3255,3223), (3260,2767), (3263,1535), (3266,798), (3267,1699), (3282,1098), (3290,1988), (3296,2093), (3306,1708), (3311,3023), (3312,2481), (3315,919), (3320,1875), (3324,1342), (3332,2474), (3341,2493), (3348,1870), (3353,2917), (3356,739), (3357,871), (3365,1673), (3369,3059), (3381,2693), (3398,2421), (3399,2959), (3401,2247), (3410,193), (3422,427), (3426,2937), (3452,2364), (3459,1255), (3467,2375), (3477,3253), (3485,783), (3489,73), (3494,1003), (3506,2407), (3509,123), (3524,539), (3531,487), (3533,985), (3536,1847), (3537,847), (3539,1643), (3545,1243), (3561,593), (3564,3341), (3570,3482), (3578,149), (3587,1791), (3593,427), (3597,937), (3609,1105), (3614,887), (3615,1535), (3618,3464), (3629,3285), (3641,2459), (3648,2379), (3657,2615), (3666,3056), (3669,1045), (3677,3015), (3686,3103), (3689,3107), (3690,2692), (3698,401), (3699,2047), (3702,2292), (3704,2503), (3708,2718), (3714,1189), (3723,1343), (3725,783), (3734,3669), (3737,293), (3747,263), (3755,827), (3759,671), (3761,1019), (3777,2821), (3779,3011), (3782,2231), (3783,3335), (3785,821), (3791,799), (3794,63), (3795,2231), (3798,2092), (3812,1435), (3821,1769), (3825,1187), (3834,2088), (3837,1235), (3851,3271), (3855,607), (3858,66), (3864,3224), (3866,307), (3867,1255), (3872,1982), (3876,3329), (3881,2461), (3887,3791), (3888,1640), (3891,1799), (3896,3377), (3920,848), (3923,779), (3924,1629), (3927,1879), (3936,2181), (3939,1159), (3941,1811), (3942,3348), (3947,2307), (3968,2964), (3969,653), (3975,3223), (3980,926), (3989,3385), (3996,3846), (4004,3722), (4005,295), (4017,743), (4020,2780), (4046,2039), (4053,3251), (4058,3371), (4059,3151), (4061,2773), (4062,1576), (4064,1435), (4067,3243), (4068,590), (4073,2597), (4091,3883), (4097,567), (4100,2737), (4101,2483), (4106,947), (4113,2231), (4115,643), (4118,377), (4139,3319), (4142,2821), (4148,889), (4149,1909), (4154,1620), (4160,3499), (4169,1309), (4179,1063), (4184,3667), (4188,452), (4197,3167), (4200,1198), (4203,427), (4211,1939), (4214,1227), (4218,3115), (4221,503), (4226,318), (4229,1427), (4232,2198), (4235,695), (4238,3444), (4241,977), (4245,835), (4247,3679), (4256,551), (4257,2287), (4259,3195), (4262,1970), (4263,1775), (4265,1489), (4266,249), (4269,2543), (4275,779), (4277,4091), (4281,3961), (4289,2387), (4293,4085), (4298,1166), (4301,1021), (4304,4116), (4307,3347), (4316,2149), (4325,461), (4332,1277), (4334,1935), (4337,2585), (4353,911), (4355,2039), (4356,2452), (4358,753), (4359,2407), (4361,1049), (4365,433), (4368,3505), (4370,2113), (4376,863), (4377,2311), (4395,4339), (4400,987), (4401,2993), (4403,3651), (4409,1781), (4418,2735), (4424,276), (4433,4319), (4436,4003), (4437,377), (4449,3119), (4451,399), (4454,240), (4455,2527), (4460,2165), (4463,1167), (4478,2304), (4481,3209), (4484,3163), (4485,149), (4490,2162), (4493,3767), (4499,319), (4505,427), (4506,1404), (4517,587), (4521,4141), (4526,3689), (4535,351), (4538,2917), (4542,3615), (4547,1743), (4548,2601), (4553,439), (4556,3509), (4559,1839), (4560,1397), (4565,4149), (4568,651), (4569,1061), (4571,2687), (4578,1734), (4580,846), (4581,541), (4586,1976), (4589,2197), (4593,503), (4608,2017), (4617,203), (4619,1455), (4629,3797), (4638,1778), (4641,4327), (4643,2135), (4646,1638), (4649,3149), (4656,244), (4658,1813), (4686,1206), (4700,1007), (4703,2559), (4707,1399), (4709,1753), (4713,3883), (4725,3079), (4728,1648), (4730,3371), (4733,3719), (4736,339), (4739,3115), (4740,1004), (4745,555), (4746,2662), (4748,824), (4752,240), (4757,35), (4761,3671), (4769,2495), (4773,2183), (4776,3427), (4784,270), (4787,4495), (4793,217), (4796,239), (4802,3989), (4805,715), (4818,2755), (4820,1099), (4821,2351), (4826,1653), (4829,4721), (4833,3359), (4835,879), (4851,511), (4853,1319), (4854,4423), (4856,4477), (4878,1071), (4881,1373), (4883,739), (4884,748), (4886,327), (4898,3849), (4905,725), (4908,232), (4917,1399), (4919,3951), (4925,2659), (4929,4297), (4931,311), (4947,547), (4952,4455), (4955,831), (4958,1553), (4961,153), (4964,4205), (4977,3875), (4989,359), (5001,3403), (5006,2591), (5009,2893), (5018,3570), (5019,391), (5025,1025), (5027,783), (5030,3686), (5037,1261), (5045,3219), (5061,3523), (5063,527), (5064,1266), (5066,4099), (5073,989), (5076,402), (5084,3252), (5093,1239), (5096,2039), (5097,4675), (5099,1055), (5102,2082), (5108,4176), (5114,335), (5121,3103), (5123,2935), (5126,2802), (5135,1999), (5139,2803), (5148,4711), (5153,2893), (5159,4799), (5165,3333), (5168,4911), (5180,4565), (5186,3548), (5187,851), (5189,4755), (5193,4717), (5195,3739), (5201,2637), (5211,1687), (5219,1391), (5226,3709), (5229,3577), (5232,4567), (5235,1007), (5237,5051), (5243,3743), (5249,1401), (5250,179), (5252,1485), (5259,1775), (5262,267), (5265,4423), (5273,5255), (5276,3979), (5285,655), (5294,2927), (5298,4206), (5300,559), (5316,5302), (5331,1939), (5333,3187), (5337,5323), (5339,2411), (5345,3893), (5348,3908), (5351,4159), (5363,2551), (5373,205), (5381,2487), (5382,4234), (5384,4317), (5388,4150), (5399,3143), (5406,1736), (5408,4602), (5409,293), (5411,1979), (5421,5263), (5433,5203), (5438,2693), (5441,5171), (5445,4349), (5453,3793), (5457,5375), (5466,1436), (5468,167), (5474,4931), (5477,5285), (5481,3853), (5483,2603), (5487,367), (5489,4013), (5493,2501), (5496,3928), (5498,1819), (5505,3371), (5520,2747), (5526,3918), (5529,2483), (5531,3587), (5535,2911), (5544,4507), (5553,5411), (5561,563), (5565,2341), (5573,1501), (5576,2952), (5589,5047), (5619,623), (5622,2668), (5625,13), (5639,4959), (5651,2123), (5658,1435), (5660,5359), (5682,4722), (5694,758), (5697,1241), (5717,3561), (5720,3433), (5721,4327), (5726,3758), (5733,3277), (5741,717), (5747,263), (5750,1710), (5754,2301), (5756,5323), (5763,203), (5765,2733), (5769,4387), (5774,488), (5777,4469), (5781,5183), (5786,5459), (5795,4567), (5807,5567), (5816,5293), (5822,4983), (5826,2469), (5829,3895), (5837,2343), (5841,5839), (5843,31), (5847,2143), (5874,3064), (5892,5790), (5921,3513), (5925,2309), (5927,4679), (5928,5111), (5934,2729), (5936,3221), (5937,3047), (5943,5255), (5945,3645), (5948,5712), (5955,2423), (5957,5593), (5958,2628), (5963,795), (5964,1648), (5966,3183), (5969,4705), (5975,439), (5978,5111), (5981,3719), (5982,2593), (5987,947), (5991,3119), (5993,5625), (5996,728), (6000,5672), (6006,1988), (6017,821), (6018,421), (6023,4911), (6026,669), (6027,1567), (6032,5094), (6039,2879), (6041,251), (6048,1605), (6056,3167), (6066,2939), (6083,1155), (6092,5461), (6096,548), (6098,1831), (6101,3433), (6104,485), (6107,779), (6119,2159), (6126,1474), (6129,2177), (6131,1543), (6132,961), (6140,2279), (6141,3521), (6155,3267), (6164,884), (6173,4181), (6177,2921), (6186,2566), (6188,773), (6189,6119), (6194,409), (6200,3455), (6225,181), (6227,683), (6230,5977), (6234,338), (6236,1063), (6237,3955), (6239,1279), (6240,4833), (6242,1622), (6249,3115), (6266,2123), (6269,253), (6281,173), (6284,3582), (6299,5987), (6320,3469), (6324,2989), (6330,1691), (6350,2015), (6354,1427), (6359,1295), (6366,671), (6371,5127), (6372,3430), (6374,3857), (6375,2567), (6377,2893), (6383,319), (6389,3317), (6393,4663), (6395,5707), (6396,5848), (6402,4246), (6405,1427), (6407,6287), (6414,1080), (6416,3267), (6417,5273), (6420,5311), (6423,3823), (6429,6191), (6432,5852), (6434,5553), (6435,3239), (6437,4253), (6449,2083), (6452,1760), (6455,3055), (6459,1891), (6473,157), (6482,1452), (6486,2018), (6489,5305), (6500,4487), (6501,3589), (6506,2789), (6507,2171), (6513,5891), (6522,6289), (6525,6431), (6536,3417), (6546,3334), (6548,3091), (6557,1385), (6558,6404), (6570,5538), (6572,2161), (6581,6199), (6584,1013), (6585,5383), (6587,107), (6588,240), (6596,1089), (6605,1653), (6627,827), (6633,1747), (6635,415), (6651,1087), (6659,1743), (6660,105), (6675,607), (6689,6613), (6690,338), (6701,4249), (6707,2887), (6710,624), (6711,6143), (6714,3505), (6740,2568), (6741,2047), (6743,6503), (6749,6339), (6750,452), (6758,4621), (6759,4319), (6761,2979), (6762,881), (6764,2906), (6767,3823), (6785,5979), (6788,2600), (6821,4921), (6822,5648), (6825,2347), (6827,3583), (6828,1702), (6830,4495), (6831,3791), (6833,1791), (6836,5581), (6837,4441), (6846,1433), (6849,2935), (6852,746), (6860,4998), (6866,6699), (6867,5755), (6873,1867), (6876,549), (6887,6103), (6888,1856), (6893,1607), (6896,6092), (6932,6859), (6935,6183), (6936,6933), (6938,2425), (6950,2226), (6956,2679), (6974,4373), (6977,6947), (6978,3676), (6981,1789), (6989,2783), (6990,5726), (6999,2191), (7001,3729), (7005,4973), (7008,1268), (7010,4650), (7013,5461), (7014,1874), (7022,3512), (7028,3236), (7029,6775), (7043,531), (7049,1829), (7050,6471), (7053,2135), (7061,6047), (7068,145), (7077,5869), (7079,4879), (7080,4350), (7082,6325), (7085,5737), (7088,4958), (7092,2147), (7094,3489), (7098,1036), (7112,4171), (7115,3739), (7119,1391), (7136,1957), (7139,3823), (7145,2567), (7146,6039), (7152,6245), (7163,1047), (7169,1103), (7170,5173), (7173,6703), (7178,2785), (7179,1723), (7181,4003), (7185,3089), (7190,4442), (7193,4513), (7202,6959), (7206,2802), (7212,4390), (7217,2473), (7218,472), (7220,2180), (7224,2888), (7226,6823), (7227,6439), (7229,6423), (7232,143), (7235,1851), (7236,2682), (7241,1817), (7260,862), (7262,6869), (7272,1805), (7290,4459), (7296,5549), (7304,3072), (7307,3403), (7316,6569), (7323,2407), (7325,3513), (7326,6493), (7329,2485), (7343,3359), (7344,825), (7346,1169), (7347,1135), (7349,4937), (7355,6907), (7359,6335), (7361,3143), (7364,2372), (7367,6047), (7368,872), (7374,1446), (7376,7242), (7379,2591), (7380,4138), (7389,227), (7400,135), (7409,2345), (7412,362), (7415,3775), (7428,302), (7434,2383), (7439,2991), (7442,1333), (7445,4595), (7451,6343), (7452,7345), (7454,561), (7458,527), (7467,4543), (7484,5043), (7487,6143), (7488,3785), (7490,1129), (7491,4751), (7497,4429), (7502,6000), (7506,5374), (7508,3019), (7509,3445), (7521,3781), (7529,1521), (7550,1046), (7574,6115), (7577,2725), (7587,6895), (7589,971), (7593,7421), (7596,6414), (7598,4201), (7610,4339), (7613,4217), (7620,3856), (7629,1261), (7640,7026), (7641,4549), (7643,1383), (7653,245), (7656,3296), (7658,5670), (7667,5515), (7674,5379), (7676,2113), (7677,5791), (7683,7403), (7686,3268), (7689,2101), (7700,2772), (7701,7237), (7707,2503), (7710,4647), (7713,2947), (7719,1879), (7728,2847), (7730,1644), (7731,5543), (7736,872), (7739,2203), (7742,4370), (7745,3975), (7757,3277), (7760,329), (7761,793), (7775,3455), (7784,4883), (7785,4667), (7787,1431), (7797,6445), (7803,4243), (7806,2363), (7808,5462), (7809,1871), (7817,4945), (7824,2168), (7826,6422), (7841,1607), (7845,2599), (7847,679), (7850,984), (7853,2695), (7856,686), (7857,875), (7869,4271), (7872,3167), (7875,5315), (7880,6029), (7883,2903), (7896,3174), (7904,500), (7905,6427), (7907,1027), (7914,2253), (7925,3861), (7929,749), (7937,1637), (7967,6463), (7977,1313), (7982,7069), (8000,938), (8001,2533), (8006,2522), (8009,2515), (8013,593), (8033,1977), (8036,1388), (8042,174), (8046,5769), (8051,6439), (8057,1673), (8064,69), (8066,4572), (8072,7785), (8073,3097), (8082,1410), (8087,687), (8093,7397), (8096,2307), (8097,5149), (8102,6425), (8105,1099), (8114,6689), (8117,735), (8120,2791), (8121,6131), (8123,7295), (8130,558), (8133,4691), (8139,7915), (8141,357), (8144,1459), (8153,4011), (8156,4421), (8162,2570), (8166,2318), (8184,4743), (8186,6811), (8187,3079), (8193,175), (8211,7963), (8213,2681), (8220,6964), (8231,8007), (8235,3583), (8237,5821), (8241,4571), (8246,2501), (8250,2015), (8256,5007), (8259,755), (8270,2669), (8276,5069), (8283,6419), (8297,4893), (8301,997), (8307,2795), (8309,2499), (8312,2219), (8315,6991), (8327,5335), (8339,6243), (8342,902), (8348,613), (8352,1263), (8354,2099), (8360,4410), (8361,5351), (8370,2376), (8372,817), (8375,7919), (8384,5201), (8385,4105), (8402,1963), (8420,8052), (8421,6067), (8441,7277), (8444,6443), (8448,5731), (8457,1811), (8460,6940), (8462,4016), (8466,5958), (8469,37), (8477,8003), (8480,1915), (8499,1555), (8504,122), (8508,2812), (8513,1137), (8525,5037), (8526,442), (8537,669), (8541,5857), (8543,7839), (8544,1658), (8549,2021), (8558,7677), (8561,6347), (8562,7486), (8564,507), (8573,5693), (8574,6126), (8577,2089), (8579,751), (8586,8246), (8592,1383), (8595,4255), (8600,4293), (8601,5497), (8604,7233), (8612,6738), (8613,1381), (8619,6223), (8622,7181), (8625,985), (8627,7883), (8631,2399), (8634,4076), (8636,3926), (8652,8280), (8657,2415), (8660,4609), (8661,4261), (8667,8507), (8669,1161), (8673,2305), (8685,1187), (8688,6464), (8691,7751), (8693,7875), (8697,3769), (8706,6124), (8717,4015), (8721,2119), (8727,7943), (8738,5516), (8741,5289), (8744,6537), (8753,383), (8756,6413), (8757,4397), (8763,623), (8768,6371), (8771,8471), (8774,5369), (8786,4122), (8805,2383), (8813,6757), (8825,4421), (8828,2642), (8835,7867), (8837,1407), (8840,2666), (8841,2233), (8843,1595), (8846,5768), (8847,6895), (8858,563), (8864,1697), (8865,1495), (8868,6236), (8873,5557), (8880,7871), (8885,1543), (8891,5819), (8892,8199), (8894,8852), (8897,6195), (8907,6727), (8909,5857), (8915,5223), (8933,701), (8937,5417), (8939,4427), (8945,8575), (8955,883), (8961,4153), (8972,8780), (8978,1467), (8993,591), (9003,8819), (9005,8553), (9011,4599), (9014,1250), (9015,7055), (9018,7367), (9020,3867), (9024,5566), (9029,7821), (9041,8657), (9044,480), (9045,4205), (9053,175), (9054,7918), (9059,1923), (9083,2007), (9089,4225), (9090,1328), (9093,2791), (9095,7263), (9099,115), (9101,8231), (9102,622), (9107,4219), (9110,2605), (9111,6023), (9114,1784), (9120,4465), (9126,9111), (9128,3626), (9132,844), (9140,3798), (9143,4759), (9144,53), (9146,6919), (9147,8935), (9150,3919), (9168,971), (9173,4219), (9176,2947), (9182,3721), (9185,6435), (9186,7994), (9189,8377), (9203,7587), (9213,6311), (9219,6791), (9225,233), (9230,878), (9237,157), (9246,9129), (9248,8531), (9254,6786), (9264,8126), (9270,812), (9275,9123), (9279,895), (9281,1753), (9285,9067), (9288,2491), (9291,2323), (9293,6335), (9294,9126), (9300,724), (9311,5951), (9317,3867), (9320,6907), (9321,4283), (9353,7267), (9360,1702), (9362,3594), (9366,5698), (9368,2423), (9369,1225), (9378,5258), (9380,4370), (9387,4403), (9393,1985), (9401,857), (9402,6521), (9405,8407), (9408,8010), (9411,2471), (9413,879), (9417,7159), (9419,675), (9428,7158), (9429,9017), (9434,1154), (9435,2171), (9443,6471), (9464,4601), (9474,7370), (9477,2345), (9480,6171), (9491,1347), (9500,5303), (9506,2372), (9510,4548), (9512,4022), (9522,9489), (9525,1843), (9534,2670), (9537,7513), (9543,2863), (9545,6737), (9551,703), (9558,3878), (9563,6039), (9570,7703), (9573,1955), (9575,4335), (9578,4385), (9579,1403), (9584,1033), (9585,4285), (9591,6239), (9600,5725), (9603,7111), (9618,838), (9620,8705), (9623,6799), (9627,2843), (9632,2546), (9636,689), (9641,4453), (9645,2095), (9653,6621), (9656,7777), (9660,8459), (9665,6761), (9666,1828), (9668,8046), (9675,3127), (9684,5344), (9687,3583), (9689,5539), (9698,8604), (9701,791), (9704,1526), (9710,5847), (9714,1707), (9716,1533), (9726,2468), (9732,910), (9741,3821), (9753,5515), (9755,8455), (9758,1159), (9761,7113), (9767,5103), (9771,67), (9773,1181), (9774,7092), (9794,6633), (9797,3635), (9806,5637), (9809,6909), (9813,1453), (9816,619), (9818,2816), (9819,6695), (9827,6751), (9830,6111), (9834,2639), (9843,8219), (9849,2933), (9851,6567), (9861,2053), (9863,855), (9864,3981), (9867,5239), (9878,7799), (9894,9519), (9896,5103), (9899,3219), (9902,8472), (9905,3785), (9915,895), (9917,7429), (9918,9568), (9920,3863), (9933,5663), (9950,3469), (9966,834), (9981,9209), (9998,5682), (10002,2872), ...

3. Primes composed of all ones except for two zeros. The number (10^a-1)/9-10^b-10^c is an ænlic prime for (a,b,c) = (12,9,1), (24,19,11), (384,353,225), ... If there are more ænlic primes of this form, then a > 2050.

I'm not aware of any ænlic primes composed of all ones except for three (or more) zeros.

Tuesday, January 22

Long time comin'


I first queried The Mudcat Café about this piece of music in August 1999. The fact that I had been unable since then to elicit a credible lead [see the final paragraph in my blog here, the Google+ posts here and (a reprise of the blog item) here] has been a thorn in my side! This evening I was going through some random old-time favourites in my music library and I thought again to look for it. Google showed me the Mudcat link and I went there to see how long ago I had posted my question. As I advanced the thread I reached a January 2017 guest posting with the above YouTube link.

To be fair, Anderson's voice wasn't exactly how my mind had recreated the memory of it, but there was little doubt that this was the song that I had last heard on the radio circa 1970 — almost fifty years ago. The YouTube audio had been up since September 2013. Three-plus years for someone to link it to the Mudcat thread and two more years for me to notice it.

Over the years I had convinced myself that I would never know the artist/song before I died. But my faith in the internet has been somewhat restored and I can cross this difficult things-to-find task off my bucket list.

Thursday, January 17

My 200th Leyland prime find


In August 2016 I mentioned my 100th Leyland prime find. This morning I found my 200th. The graph (click on it for a better view) shows the number of decimal digits of those 200 primes as a function of the discovery date (the calendar years mark the start of those years). One may note the periods of relative inactivity where I was burdening my processors with other tasks. The upper-right plateau constitutes my "long compute" which will come in at just under two years when it's done in April of this year.

Sunday, December 30

Reinforcements


A month ago I acquired three Mac minis to add to my arsenal of large-prime-searching computers. I've never had a Mac mini before so I wasn't at all sure what I might be getting into. In particular, their 3.2 GHz clock speed was less than the 3.4 GHz on my six-year-old iMac and the 3.5 GHz on my five-year-old Mac Pro.

I needn't have worried. My Leyland-prime search is organized into bundles of 88670 Leyland numbers per core. At the current ~101500-digit number size, it takes my Mac Pro about 43 days to cover that space. My old iMac is actually a couple of days faster — even though (as my main machine) I'm running a handful of other things on it at any given time (which slows it down a bit). I am now near the end of a run on two of the Mac minis which will (tomorrow) have covered the space in under 29 days!

Monday, December 24

Go stuff yourself!

When Eric Angelini and Jean-Marc Falcoz submitted to the OEIS on December 3 their stuffable numbers sequence (see my Stuffing post last month), they also submitted a self-stuffable numbers sequence to accompany it. The same day, to the latter I added a 41-term "b-file". On December 6, John Mason replaced that with an 84-term file. Although I had by then calculated more than 41 terms, I would never have reached 84 terms (< 10^15) with my clumsy brute-force effort. Looking at Mason's 84 terms, I noticed (and soon proved to myself) that the numbers 103008000000, 1031008000000, 10311008000000, 103111008000000, ... were the beginning of an infinite family of self-stuffable numbers, an unexpected and utterly delightful emergence.

I rewrote my program for generating self-stuffables and managed 95 terms (< 10^17) by December 12, on which day I submitted my observation about the 103008000000 family. One week later — still waiting for that comment to be approved — I decided to advertise it on the Sequence Fanatics Discussion list. Although I had noted that it was an infinite family, John Mason felt it important to point out that he had extended (but not yet submitted) his b-file and had found 1031111008000000 and 10311111008000000. In order to give John a heads-up, I emailed him privately and told him that I was four days away from finishing my 18-digit terms (<10^18), adding at least three dozen terms (which I had by then calculated) to the 95 terms we had both reached. I asked John if he was working on the 18-digit terms as well. He replied that he wasn't. If he had been, I would have offered to wait until he was done so that we might co-author the extended b-file together.

Two days later (December 21), John posted to the Sequence Fanatics Discussion list that he had calculated the 18-digit terms and had submitted his new 134-term b-file. Interestingly, he left my count at 41. Ouch! Mason had the audacity to complain that he was looking for efficiencies in his program as it had taken him all of eight hours to run. Mine was taking eleven days! To add insult to injury, on December 22 the iMac on which my program was running suffered a kernel panic and — one day short — I lost the completion of it.

2019 update: Rewriting my program yet again, I finished the 18-digit terms on January 1. Lars Blomberg suggested 165 terms (<10^22) on January 2 but it was noticed by Ray Chandler that at least one term (21021021021021021021) was missing! John Mason uploaded 145 terms (<10^20) on January 14 that included Blomberg's missing number. I finished the 19-digit terms on January 18. Too little, too late.

Monday, November 26

Stuffing

Eric Angelini ponders stuffable numbers in his November 24 blog posting (the second article here). If the digits of an n-digit (base ten) number are d1, d2, d3, ... d(n-1), d(n), then d1 new digits are inserted after d1, d2 new digits after d2, d3 new digits after d3, ... and d(n-1) new digits after d(n-1). But they can't just be any digits. The concatenation of all the new digits (call it stuff) must be such that the newly created integer (stuffed with all the inserted digits) is a multiple of this stuff.

Eric gives the example of 2018 which is stuffable because 2130168 is a multiple of 136. In addition to 136, 2018 is stuffable by 143, 154, 319, 418, and 946. In contrast, 2019 is not stuffable at all. We expect stuff to not begin with a leading zero, so the new digits after d1 will not begin with a zero. However, the new digits after d2, d3, ... d(n-1) might begin with one or more leading zeros. They might even be all zeros.

Here's another example. 111111111 is stuffable by 13546599, 15969681, 19019019, 57057057, 70060833, 71071071, 77077077, and 89302941:

11131514161519191 is a multiple of 13546599
11151916191618111 is a multiple of 15969681
11191011191011191 is a multiple of 19019019
15171015171015171 is a multiple of 57057057
17101016101813131 is a multiple of 70060833
17111017111017111 is a multiple of 71071071
17171017171017171 is a multiple of 77077077
18191310121914111 is a multiple of 89302941

If we use ten ones (1111111111) instead of nine, that number is stuffable by only 101010101 and 468397877:

1110111011101110111 is a multiple of 101010101
1416181319171817171 is a multiple of 468397877

The presence of larger digits in all but the final (units) position severely restricts one's ability to brute-force solutions, which is why Jean-Marc Falcoz's table (in Eric's blog) of least solution only goes up to 90. My own table of all solutions goes up to 379, followed by some examples >1000 and >2000.

Sunday, October 28

What's so special about 102735?

There are 8877691 nonnegative integers with distinct decimal digits. Last week, Eric Angelini suggested a procedure for numbers that do contain duplicated digits to be reduced to terms of this sequence simply by erasing all digits that appear more than once and bunching up the digits that remain. Leading zeros are of course ignored and we will posit that integers that disappear entirely become 0, so as to remain in the aforementioned A010784 set.

Iterating some starting integer by successive multiplication of 2, 3, 4, 5, 6, etc. and reducing at each turn, if necessary (as per the above), one hopes for a large number of turns before reaching 0. For example, starting with 24603:

2 * 24603 = 49206
3 * 49206 = [147618] => 4768
4 * 4768 = 19072
5 * 19072 = 95360
6 * 95360 = 572160
7 * 572160 = [4005120] => 4512
8 * 4512 = [36096] => 309
9 * 309 = 2781
10 * 2781 = 27810
11 * 27810 = [305910] => 3591
12 * 3591 = 43092
13 * 43092 = [560196] => 5019
14 * 5019 = [70266] => 702
15 * 702 = [10530] => 153
16 * 153 = [2448] => 28
17 * 28 = 476
18 * 476 = [8568] => 56
19 * 56 = 1064
20 * 1064 = [21280] => 180
21 * 180 = 3780
22 * 3780 = 83160
23 * 83160 = [1912680] => 92680
24 * 92680 = [2224320] => 430
25 * 430 = [10750] => 175
26 * 175 = [4550] => 40
27 * 40 = [1080] => 18
28 * 18 = 504
29 * 504 = [14616] => 4
30 * 4 = 120
31 * 120 = 3720
32 * 3720 = [119040] => 94
33 * 94 = 3102
34 * 3102 = 105468
35 * 105468 = [3691380] => 69180
36 * 69180 = [2490480] => 298
37 * 298 = [11026] => 26
38 * 26 = [988] => 9
39 * 9 = 351
40 * 351 = [14040] => 1
41 * 1 = 41
42 * 41 = [1722] => 17
43 * 17 = 731
44 * 731 = 32164
45 * 32164 = [1447380] => 17380
46 * 17380 = [799480] => 7480
47 * 7480 = [351560] => 3160
48 * 3160 = [151680] => 5680
49 * 5680 = [278320] => 7830
50 * 7830 = [391500] => 3915
51 * 3915 = [199665] => 15
52 * 15 = 780
53 * 780 = [41340] => 130
54 * 130 = [7020] => 72
55 * 72 = 3960
56 * 3960 = [221760] => 1760
57 * 1760 = [100320] => 132
58 * 132 = [7656] => 75
59 * 75 = [4425] => 25
60 * 25 = [1500] => 15
61 * 15 = 915
62 * 915 = 56730
63 * 56730 = [3573990] => 570
64 * 570 = 36480
65 * 36480 = [2371200] => 371
66 * 371 = [24486] => 286
67 * 286 = [19162] => 962
68 * 962 = [65416] => 541
69 * 541 = [37329] => 729
70 * 729 = [51030] => 513
71 * 513 = [36423] => 642
72 * 642 = [46224] => 6
73 * 6 = 438
74 * 438 = [32412] => 341
75 * 341 = [25575] => 27
76 * 27 = [2052] => 5
77 * 5 = 385
78 * 385 = [30030] => 0
79 * 0 = 0

Here, numbers that contain duplicated digits are shown in [square brackets] => followed by the reduced number. If we replace the iterated multiplication of successive integers with the iterated multiplication of successive primes, we can go much further. 102735 is the smallest integer with distinct decimal digits that generates the longest such chain:

2 * 102735 = [205470] => 2547
3 * 2547 = 7641
5 * 7641 = 38205
7 * 38205 = 267435
11 * 267435 = 2941785
13 * 2941785 = [38243205] => 8405
17 * 8405 = [142885] => 1425
19 * 1425 = [27075] => 205
23 * 205 = 4715
29 * 4715 = [136735] => 1675
31 * 1675 = [51925] => 192
37 * 192 = 7104
41 * 7104 = [291264] => 9164
43 * 9164 = 394052
47 * 394052 = [18520444] => 18520
53 * 18520 = 981560
59 * 981560 = [57912040] => 579124
61 * 579124 = [35326564] => 24
67 * 24 = 1608
71 * 1608 = [114168] => 468
73 * 468 = [34164] => 316
79 * 316 = [24964] => 296
83 * 296 = 24568
89 * 24568 = [2186552] => 186
97 * 186 = 18042
101 * 18042 = [1822242] => 184
103 * 184 = 18952
107 * 18952 = [2027864] => 7864
109 * 7864 = [857176] => 8516
113 * 8516 = 962308
127 * 962308 = [122213116] => 36
131 * 36 = 4716
137 * 4716 = [646092] => 4092
139 * 4092 = [568788] => 567
149 * 567 = [84483] => 3
151 * 3 = 453
157 * 453 = [71121] => 72
163 * 72 = [11736] => 736
167 * 736 = [122912] => 9
173 * 9 = [1557] => 17
179 * 17 = [3043] => 4
181 * 4 = 724
191 * 724 = [138284] => 1324
193 * 1324 = [255532] => 3
197 * 3 = 591
199 * 591 = [117609] => 7609
211 * 7609 = [1605499] => 16054
223 * 16054 = [3580042] => 35842
227 * 35842 = [8136134] => 864
229 * 864 = 197856
233 * 197856 = [46100448] => 618
239 * 618 = [147702] => 1402
241 * 1402 = [337882] => 72
251 * 72 = 18072
257 * 18072 = [4644504] => 650
263 * 650 = [170950] => 1795
269 * 1795 = [482855] => 42
271 * 42 = [11382] => 382
277 * 382 = [105814] => 584
281 * 584 = [164104] => 60
283 * 60 = 16980
293 * 16980 = [4975140] => 97510
307 * 97510 = [29935570] => 2370
311 * 2370 = [737070] => 3
313 * 3 = [939] => 3
317 * 3 = 951
331 * 951 = [314781] => 3478
337 * 3478 = [1172086] => 72086
347 * 72086 = [25013842] => 501384
349 * 501384 = [174983016] => 7498306
353 * 7498306 = [2646902018] => 4918
359 * 4918 = [1765562] => 172
367 * 172 = 63124
373 * 63124 = [23545252] => 34
379 * 34 = [12886] => 126
383 * 126 = [48258] => 425
389 * 425 = [165325] => 1632
397 * 1632 = [647904] => 6790
401 * 6790 = [2722790] => 90
409 * 90 = 36810
419 * 36810 = [15423390] => 154290
421 * 154290 = [64956090] => 45
431 * 45 = [19395] => 135
433 * 135 = [58455] => 84
439 * 84 = [36876] => 387
443 * 387 = [171441] => 7
449 * 7 = [3143] => 14
457 * 14 = 6398
461 * 6398 = [2949478] => 278
463 * 278 = [128714] => 2874
467 * 2874 = [1342158] => 34258
479 * 34258 = 16409582
487 * 16409582 = [7991466434] => 713
491 * 713 = [350083] => 58
499 * 58 = [28942] => 894
503 * 894 = [449682] => 9682
509 * 9682 = [4928138] => 49213
521 * 49213 = [25639973] => 2567
523 * 2567 = [1342541] => 325
541 * 325 = [175825] => 1782
547 * 1782 = [974754] => 95
557 * 95 = [52915] => 291
563 * 291 = [163833] => 168
569 * 168 = [95592] => 2
571 * 2 = [1142] => 42
577 * 42 = [24234] => 3
587 * 3 = [1761] => 76
593 * 76 = 45068
599 * 45068 = [26995732] => 6573
601 * 6573 = [3950373] => 9507
607 * 9507 = [5770749] => 5049
613 * 5049 = [3095037] => 957
617 * 957 = [590469] => 5046
619 * 5046 = [3123474] => 127
631 * 127 = 80137
641 * 80137 = [51367817] => 5368
643 * 5368 = [3451624] => 35162
647 * 35162 = [22749814] => 7981
653 * 7981 = [5211593] => 293
659 * 293 = 193087
661 * 193087 = [127630507] => 12635
673 * 12635 = [8503355] => 80
677 * 80 = 54160
683 * 54160 = [36991280] => 361280
691 * 361280 = [249644480] => 29680
701 * 29680 = [20805680] => 256
709 * 256 = [181504] => 8504
719 * 8504 = [6114376] => 437
727 * 437 = [317699] => 3176
733 * 3176 = [2328008] => 3
739 * 3 = [2217] => 17
743 * 17 = [12631] => 263
751 * 263 = [197513] => 9753
757 * 9753 = [7383021] => 78021
761 * 78021 = [59373981] => 5781
769 * 5781 = [4445589] => 89
773 * 89 = [68797] => 689
787 * 689 = [542243] => 53
797 * 53 = [42241] => 1
809 * 1 = 809
811 * 809 = [656099] => 50
821 * 50 = [41050] => 415
823 * 415 = [341545] => 31
827 * 31 = 25637
829 * 25637 = [21253073] => 1507
839 * 1507 = [1264373] => 12647
853 * 12647 = [10787891] => 9
857 * 9 = [7713] => 13
859 * 13 = [11167] => 67
863 * 67 = 57821
877 * 57821 = [50709017] => 591
881 * 591 = 520671
883 * 520671 = [459752493] => 723
887 * 723 = [641301] => 6430
907 * 6430 = [5832010] => 58321
911 * 58321 = [53130431] => 504
919 * 504 = [463176] => 4317
929 * 4317 = [4010493] => 193
937 * 193 = [180841] => 4
941 * 4 = 3764
947 * 3764 = [3564508] => 36408
953 * 36408 = [34696824] => 3982
967 * 3982 = [3850594] => 38094
971 * 38094 = [36989274] => 368274
977 * 368274 = [359803698] => 506
983 * 506 = [497398] => 4738
991 * 4738 = [4695358] => 46938
997 * 46938 = [46797186] => 4918
1009 * 4918 = [4962262] => 49
1013 * 49 = 49637
1019 * 49637 = [50580103] => 813
1021 * 813 = [830073] => 87
1031 * 87 = [89697] => 867
1033 * 867 = [895611] => 8956
1039 * 8956 = 9305284
1049 * 9305284 = [9761242916] => 74
1051 * 74 = [77774] => 4
1061 * 4 = [4244] => 2
1063 * 2 = [2126] => 16
1069 * 16 = [17104] => 704
1087 * 704 = 765248
1091 * 765248 = [834885568] => 346
1093 * 346 = [378178] => 31
1097 * 31 = [34007] => 347
1103 * 347 = 382741
1109 * 382741 = [424459769] => 2576
1117 * 2576 = [2877392] => 839
1123 * 839 = [942197] => 4217
1129 * 4217 = [4760993] => 47603
1151 * 47603 = [54791053] => 479103
1153 * 479103 = [552405759] => 24079
1163 * 24079 = [28003877] => 23
1171 * 23 = [26933] => 269
1181 * 269 = 317689
1187 * 317689 = [377096843] => 9684
1193 * 9684 = [11553012] => 302
1201 * 302 = [362702] => 3670
1213 * 3670 = [4451710] => 570
1217 * 570 = [693690] => 30
1223 * 30 = [36690] => 390
1229 * 390 = 479310
1231 * 479310 = [590030610] => 59361
1237 * 59361 = [73429557] => 3429
1249 * 3429 = [4282821] => 41
1259 * 41 = [51619] => 569
1277 * 569 = [726613] => 7213
1279 * 7213 = [9225427] => 9547
1283 * 9547 = [12248801] => 40
1289 * 40 = [51560] => 160
1291 * 160 = [206560] => 25
1297 * 25 = [32425] => 345
1301 * 345 = [448845] => 5
1303 * 5 = [6515] => 61
1307 * 61 = [79727] => 92
1319 * 92 = [121348] => 2348
1321 * 2348 = [3101708] => 378
1327 * 378 = [501606] => 51
1361 * 51 = [69411] => 694
1367 * 694 = [948698] => 46
1373 * 46 = 63158
1381 * 63158 = [87221198] => 79
1399 * 79 = [110521] => 52
1409 * 52 = 73268
1423 * 73268 = [104260364] => 123
1427 * 123 = [175521] => 72
1429 * 72 = [102888] => 102
1433 * 102 = [146166] => 4
1439 * 4 = [5756] => 76
1447 * 76 = [109972] => 1072
1451 * 1072 = [1555472] => 1472
1453 * 1472 = [2138816] => 236
1459 * 236 = [344324] => 2
1471 * 2 = [2942] => 94
1481 * 94 = [139214] => 3924
1483 * 3924 = [5819292] => 581
1487 * 581 = 863947
1489 * 863947 = [1286417083] => 264703
1493 * 264703 = [395201579] => 32017
1499 * 32017 = [47993483] => 78
1511 * 78 = [117858] => 75
1523 * 75 = [114225] => 45
1531 * 45 = [68895] => 695
1543 * 695 = 1072385
1549 * 1072385 = [1661124365] => 2435
1553 * 2435 = [3781555] => 3781
1559 * 3781 = [5894579] => 847
1567 * 847 = [1327249] => 13749
1571 * 13749 = [21599679] => 21567
1579 * 21567 = [34054293] => 529
1583 * 529 = [837407] => 8340
1597 * 8340 = [13318980] => 90
1601 * 90 = [144090] => 19
1607 * 19 = [30533] => 5
1609 * 5 = 8045
1613 * 8045 = [12976585] => 129768
1619 * 129768 = [210094392] => 143
1621 * 143 = [231803] => 2180
1627 * 2180 = [3546860] => 35480
1637 * 35480 = [58080760] => 576
1657 * 576 = [954432] => 9532
1663 * 9532 = [15851716] => 876
1667 * 876 = [1460292] => 14609
1669 * 14609 = [24382421] => 381
1693 * 381 = [645033] => 6450
1697 * 6450 = [10945650] => 1946
1699 * 1946 = [3306254] => 6254
1709 * 6254 = [10688086] => 1
1721 * 1 = [1721] => 72
1723 * 72 = 124056
1733 * 124056 = [214989048] => 210
1741 * 210 = [365610] => 3510
1747 * 3510 = [6131970] => 63970
1753 * 63970 = [112139410] => 23940
1759 * 23940 = [42110460] => 26
1777 * 26 = [46202] => 460
1783 * 460 = [820180] => 21
1787 * 21 = [37527] => 352
1789 * 352 = [629728] => 6978
1801 * 6978 = [12567378] => 125638
1811 * 125638 = [227530418] => 7530418
1823 * 7530418 = [13727952014] => 39504
1831 * 39504 = [72331824] => 7184
1847 * 7184 = [13268848] => 13264
1861 * 13264 = [24684304] => 26830
1867 * 26830 = [50091610] => 596
1871 * 596 = [1115116] => 56
1873 * 56 = [104888] => 104
1877 * 104 = 195208
1879 * 195208 = [366795832] => 79582
1889 * 79582 = [150330398] => 1598
1901 * 1598 = [3037798] => 98
1907 * 98 = [186886] => 1
1913 * 1 = [1913] => 93
1931 * 93 = 179583
1933 * 179583 = [347133939] => 471
1949 * 471 = [917979] => 1
1951 * 1 = [1951] => 95
1973 * 95 = 187435
1979 * 187435 = [370933865] => 709865
1987 * 709865 = [1410501755] => 47
1993 * 47 = 93671
1997 * 93671 = [187060987] => 169
1999 * 169 = [337831] => 781
2003 * 781 = [1564343] => 156
2011 * 156 = [313716] => 76
2017 * 76 = [153292] => 1539
2027 * 1539 = [3119553] => 9
2029 * 9 = [18261] => 826
2039 * 826 = [1684214] => 682
2053 * 682 = [1400146] => 6
2063 * 6 = 12378
2069 * 12378 = [25610082] => 5618
2081 * 5618 = [11691058] => 69058
2083 * 69058 = [143847814] => 37
2087 * 37 = [77219] => 219
2089 * 219 = [457491] => 5791
2099 * 5791 = [12155309] => 2309
2111 * 2309 = [4874299] => 872
2113 * 872 = 1842536
2129 * 1842536 = [3922759144] => 3751
2131 * 3751 = [7993381] => 781
2137 * 781 = [1668997] => 187
2141 * 187 = [400367] => 4367
2143 * 4367 = [9358481] => 93541
2153 * 93541 = [201393773] => 2019
2161 * 2019 = [4363059] => 46059
2179 * 46059 = [100362561] => 325
2203 * 325 = [715975] => 19
2207 * 19 = [41933] => 419
2213 * 419 = [927247] => 94
2221 * 94 = [208774] => 2084
2237 * 2084 = [4661908] => 41908
2239 * 41908 = [93832012] => 9801
2243 * 9801 = [21983643] => 219864
2251 * 219864 = [494913864] => 1386
2267 * 1386 = [3142062] => 31406
2269 * 31406 = [71260214] => 7604
2273 * 7604 = [17283892] => 1739
2281 * 1739 = [3966659] => 35
2287 * 35 = [80045] => 845
2293 * 845 = [1937585] => 19378
2297 * 19378 = [44511266] => 52
2309 * 52 = [120068] => 1268
2311 * 1268 = [2930348] => 29048
2333 * 29048 = [67768984] => 94
2339 * 94 = [219866] => 2198
2341 * 2198 = [5145518] => 48
2347 * 48 = [112656] => 25
2351 * 25 = [58775] => 8
2357 * 8 = [18856] => 156
2371 * 156 = [369876] => 3987
2377 * 3987 = [9477099] => 40
2381 * 40 = 95240
2383 * 95240 = [226956920] => 50
2389 * 50 = [119450] => 9450
2393 * 9450 = [22613850] => 613850
2399 * 613850 = [1472626150] => 4750
2411 * 4750 = [11452250] => 40
2417 * 40 = [96680] => 980
2423 * 980 = [2374540] => 23750
2437 * 23750 = [57878750] => 0
2441 * 0 = 0