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| maple blossoms: April 25 |
Now that Ed Pegg's recent Math-Fun suggestion is ensconced in the OEIS, I will highlight his assertion that the smallest product with a single-digit factorization is 1476395008. My idea is to enumerate a bunch of such integers by multiplying together all possible combinations of all possible powers of repdigits (of 2, 3, 4, 7, 8, 9), ignoring numbers larger than some limit. The products are then examined for having the nine digits that are not the factorization digit.
I managed to generate 2554 terms (<10^24) before running out of RAM. Michael Branicky upped this to 10000 terms (available as a b-file in OEIS A372106). Here is how things start:
I had put the fully factored 10000 terms here but Neil saw fit to add it to the OEIS.
Based on a Neil Sloane misreading of an Ed Pegg idea (link has 187511 products, 10 MB):
8596 = 2*14*307
8790 = 2*3*1465
9360 = 2*4*15*78
9380 = 2*5*14*67
9870 = 2*3*1645
10752 = 3*4*896
12780 = 4*5*639
14760 = 5*9*328
14820 = 5*39*76
15628 = 4*3907
15678 = 39*402
16038 = 27*594 = 54*297
16704 = 9*32*58
17082 = 3*5694
17820 = 36*495 = 45*396
17920 = 8*35*64
18720 = 4*5*936
19084 = 52*367
19240 = 8*37*65
20457 = 3*6819
20574 = 6*9*381
20754 = 3*6918
21658 = 7*3094
24056 = 8*31*97
24507 = 3*8169
25803 = 9*47*61
26180 = 4*7*935
26910 = 78*345
27504 = 3*9168
28156 = 4*7039
28651 = 7*4093
30296 = 7*8*541
30576 = 8*42*91
30752 = 4*8*961
31920 = 5*76*84
32760 = 8*45*91
32890 = 46*715
34902 = 6*5817
36508 = 4*9127
47320 = 8*65*91
58401 = 63*927
65128 = 7*9304
65821 = 7*9403
Neil has fast-tracked this into the OEIS.
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In addition to my own "fanciful extension" of Éric Angelini's Two identical digits effort, Jean-Marc Falcoz suggested his own variation (at the end of the blog entry): "Lexicographically earliest sequence of distinct positive terms such that [the product of adjacent terms] contains exactly 1 digit 1 (if 1 is present), 2 digits 2 (if 2 is present), 3 digits 3 (if 3 is present), ... 9 digits 9 (if 9 is present)." He presented 113 terms of the sequence but I was hungry for more.
The above plot just exceeds 1000 terms. Surprisingly, term #318 is 17 and term #319 is 13, local minima. Term #455 is 1011211671, a thus-far maximum. Possible products are given by A108571. Our indexed products are such that product #2 is term #2 multiplied by term #1 (product #1 is 1 by fiat). In the current list there is only one duplicate: product #172 = product #622 = 2423433144. If typed by their constituent digits, regardless of digit order, the number of possible types is given by A125573. Our current list realizes just 71, the number of which (sorted by product digit-length) are: 1, 1, 1, 2, 0, 0, 2, 3, 2, 4, 5, 7, 9, 11, 9, 10, 4.
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The following described sequence is a somewhat fanciful extension of Éric Angelini's latest effort, Two identical digits. In my version, the products of adjacent terms must contain a single occurrence of the three digits "666". Furthermore, those products may not contain any additional sixes. The lexicographically earliest sequence of distinct positive terms starts:
1, 666, 10, 1666, 4, 1665, 40, 1667, 400, 4165, 16, 4163, 160, 4164, 1600, 4167, 64, 1041, 640, 1042, 373, 42, 1111, 15, 444, 150, 2444, 506, 361, 988, 675, 395, 422, 203, 1422, 469, 398, 268, 1368, 173, 385, 329, 154, 303, 55, 1211, 406, 211, 79, 654, 316, 1635, 579, 616, 1082, 191, 349, 234, 285, 269, 114, 585, 849, 903, 738, 257, 2166, 251, 166, 751, 355, 723, 142, 223, 299, 534, 312, 1047, 78, 47, 780, 470, 1078, 547, 195, 188, 461, 906, 961, 694, 96, 2778, 6, 111, 60, 611, 551, 121, 146, 621, 365, 484, 427, 156, 235, 539, 94, 39, 171, 390, 940, 922, 453, 1273, 288, 926, 18, 37, 180, 370, 991, 148, 45, 1037, 450, 1480, 518, 287, 929, 61, 306, 561, 101, 165, 1010, 264, 1351, 493, 338, 286, 233, 402, 733, 773, 345, 483, 138, 157, 552, 1407, 474, 109, 1896, 879, 292, 605, 573, 242, 73, 913, 730, 1242, 537, 869, 767, 598, 267, 624, 1068, 1186, 181, 186, 1181, 302, 383, 174, 159, 419, 214, 1246, 271, 2458, 556, 1198, 367, 198, 867, 769, 997, 669, 548, 1034, 49, 34, 196, 85, 549, 340, 490, 136, 1225, 544, 1226, 87, 318, 524, 757, 273, 244, 683, 449, 438, 207, 322, 353, 1237, 1024, 1627, 84, 793, 248, 1075, 62, 43, 155, 172, 562, 1542, 123, 2542, 262, 636, 435, 613, 882, 113, 59, 452, 590, 774, 559, 477, 58, 1149, 145, 1839, 294, 339, 1294, 515, 712, 936, 178, 206, 411, 892, 747, 278, 959, 374, 459, 363, 382, 541, 308, 671, 993, 1678, 588, 1133, 802, 532, 1441, 226, 118, 387, 689, 194, 756, 1940, 1189, 485, 1374, 558, 727, 243, 686, 431, 1547, 237, 218, 948, 545, 1223, 1003, 222, 3, 1222, 30, 2220, 300, 2221, 546, 122, 153, 305, 612, 1939, 1071, 249, 589, 283, 702, 95, 807, 38, 307, 152, 1557, 538, 57, 117, 570, 1169, 714, 169, 572, 641, 182, 163, 1182, 663, 553, 522, 53, 1257, 530, 1258, 177, 258, 677, 645, 708, 758, 927, 719, 51, 366, 510, 915, 204, 866, 77, 606, 44, 2197, 440, 3106, 261, 106, 629, 265, 1044, 638, 407, 1466, 201, 466, 143, 662, 643, 648, 1028, 843, 791, 126, 529, 315, 1291, 284, 446, 71, 1446, 571, 467, 798, 835, 499, 1837, 818, 337, 999, 334, 1499, 378, 388, 1718, 1487, 603, 608, 1261, 1060, 761, 219, 414, 161, 706, 1161, 574, 259, 296, 225, 1185, 436, 1528, 1863, 358, 27, 247, 270, 2467, 404, 66, 601, 610, 765, 871, 421, 346, 684, 536, 199, 737, 904, 295, 565, 472, 1137, 586, 455, 652, 1022, 212, 739, 902, 1483, 989, 394, 489, 1362, 372, 905, 593, 281, 344, 775, 86, 31, 215, 124, 457, 938, 711, 375, 1511, 441, 582, 252, 2209, 845, 789, 376, 975, 1094, 1432, 277, 858, 439, 653, 1021, 555, 12, 1389, 48, 1388, 480, 3471, 46, 471, 184, 362, 93, 717, 298, 1717, 33, 505, 330, 808, 1283, 1377, 397, 445, 824, 2022, 577, 658, 770, 865, 771, 246, 1271, 139, 1494, 473, 141, 26, 1141, 260, 1410, 1026, 65, 564, 526, 391, 2526, 891, 187, 918, 1053, 633, 911, 622, 403, 488, 1365, 884, 1727, 965, 691, 82, 313, 820, 1313, 1117, 597, 1116, 454, 779, 599, 734, 99, 673, 990, 1683, 102, 183, 255, 732, 91, 293, 876, 1035, 644, 647, 103, 356, 468, 698, 617, 886, 231, 202, 132, 2020, 533, 125, 1333, 2, 333, 20, 833, 8, 2083, 80, 3333, 5, 1332, 50, 3332, 200, 1833, 511, 424, 393, 1391, 479, 254, 656, 2540, 853, 781, 854, 379, 1073, 2242, 1234, 443, 462, 772, 1019, 193, 962, 1193, 475, 1403, 354, 129, 517, 98, 17, 392, 170, 980, 68, 245, 272, 2449, 634, 736, 2264, 692, 342, 423, 1316, 1654, 790, 1097, 945, 705, 52, 1281, 520, 1859, 974, 359, 1856, 1329, 923, 939, 497, 1341, 609, 602, 1107, 838, 107, 2492, 729, 503, 888, 75, 889, 750, 2222, 21, ...
The region from 740000 to 890000 is detailed here:
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1938 Two Sleepy People 3:07 Hoagy Carmichael, Ella Logan
1955 Ain't That A Shame 2:25 Fats Domino
1957 Wake Up Little Susie 2:04 The Everly Brothers
1958 True Love Ways 3:02 Buddy Holly, Dick Jacobs Orchestra
1959 ('Til) I Kissed You 2:25 The Everly Brothers
1959 Handy Man 2:06 Jimmy Jones, Otis Blackwell
1959 Calendar Girl 2:40 Neil Sedaka
1960 Good Timin' 2:13 Jimmy Jones
1960 Walk, Don't Run 2:09 The Ventures
1960 Funnel Of Love 2:07 Wanda Jackson
1961 Tell Old Bill 4:24 Dave Van Ronk
1961 Runaway 2:19 Del Shannon
1962 Green Onions 2:52 Booker T. and The MG's
1962 Pipeline 2:22 The Chantays
1962 He's So Fine 1:48 The Chiffons
1962 Wolverton Mountain 2:58 Claude King
1962 Palisades Park 1:54 Freddy Cannon
1963 It's My Party 2:22 Lesley Gore
1965 Tombstone Blues 6:01 Bob Dylan
1965 Colours 2:48 Donovan
1965 These Boots Are Made For Walkin' 2:46 Nancy Sinatra
1965 The Cuckoo 3:25 Tom Rush
1967 I Am The Walrus 4:36 The Beatles
1967 (Sittin' On) The Dock Of The Bay 2:50 Otis Redding
1968 Birthday 2:43 The Beatles
1968 Ducks On A Pond 9:11 The Incredible String Band
1969 Across The Universe 3:48 The Beatles
1969 The Boxer 5:13 Simon & Garfunkel
1970 I Heard It Through The Grapevine 11:07 Creedence Clearwater Revival
1970 The Story In Your Eyes 2:57 The Moody Blues
1971 Ain't No Sunshine 2:03 Bill Withers
1973 Raised On Robbery 3:07 Joni Mitchell
1973 Cam Ye O'er Frae France 2:50 Steeleye Span
1975 Bandalabourou 6:45 Ali Farka Touré
1975 Diamonds & Rust 4:46 Joan Baez
1978 Sultans of swing 5:36 Dire Straits
1979 Planet Claire 4:37 the B-52's
1979 Echo Beach 3:40 Martha and the Muffins
1979 Another Brick In The Wall (Part II) 4:01 Pink Floyd
1980 Could You Be Loved 3:35 Bob Marley and the Wailers
1980 Mirror In The Bathroom 3:09 The English Beat
1982 Farewell to Nova Scotia 3:11 Touchstone
1983 Get The Balance Right (combination mix) 8:00 Depeche Mode
1983 Blue Monday 7:29 New Order
1983 Johnny B. Goode 4:05 Peter Tosh
1984 Ain't Necessarily So 4:43 Bronski Beat
1984 The Bottomless Lake 3:42 John Prine
1984 Come Out And Dance 4:50 Martha and the Muffins
1985 Don't Stop The Dance 4:19 Bryan Ferry
1985 Losers 3:14 Dave Van Ronk
1987 Hush Little Baby 4:44 The Horse Flies
1987 I'm Your Man 4:29 Leonard Cohen
1989 Alasdair Mhic Cholla Ghasda 2:31 Capercaillie
1991 Stand By The JAMs (12" version) 5:32 The KLF, Tammy Wynette
1992 A Night In The Mountains 8:37 Rabih Abou-Khalil
1993 Mama Sara 7:13 Farafina
1993 The River 6:31 Geoffrey Oryema
1995 All My Tears 3:42 Emmylou Harris
1995 Carnival 5:59 Natalie Merchant
1996 Sardinia Memories (After Hours) 2:31 Geoffrey Oryema
1996 Acony Bell 3:06 Gillian Welch
1996 da eye wifey 7:47 shooglenifty
1997 Night Ride Across the Caucasus 8:33 Loreena McKennitt
1998 Winter's Come And Gone 2:15 Gillian Welch
2002 Garden Tree 5:43 John Brown's Body
2003 As Time Goes By 3:49 Rod Stewart, Queen Latifah
2004 Woman King 4:21 Iron & Wine
2005 Thanks For The Memory 3:11 Rod Stewart, Roy Hargrove
2007 Dark Undercoat 4:57 Emily Jane White
2008 Roflcopter 7:21 Ott
Back in 2003 I contributed in The Mudcat Café transcribed-by-ear lyrics for Dave Van Ronk's "Losers", noting that I was "unsure about three or so words". Now, a couple of decades later, I'm finally prepared (thanks largely to Elijah Wald) to fix that problematic second verse:
I blew my wad playing seven-card-stud
I was playing for money, they was playing for blood
On the way back home the big winner got mugged
Now he's just another loser like me
Losers, losers
I got took for my whosis
That shark got crowned: He's groan bin bound
He's just another loser like me
See that kid sitting back at the bar
He's picking up a storm on a Martin guitar
That poor fool thinks he's gonna be a star
He's just another loser like me
Losers, losers
Some are raggers, some are bluesers
Makin' disco sounds in a HoJo lounge
With a bunch of other losers like me
Love has busted up this cat for sure
He's crying like a baby at his baby's door
That poor fool don't know what he's crying for
He's just another loser like me
Losers, losers
Can't say no to cruisers
When she says "When he'll be back again?"
He's just another loser like me
There's a hobo up in heaven on the golden street
He'll panhandle every angel that he'll meet
He'd hock his harp for some Sneaky Pete
He's just another loser like me
Losers, losers
Some are dopers, some are boozers
All the muscatel is down in hell
He's just another loser like me
When God appeared to Saint John Wayne
He told him "Duke, I'm a-coming again
Life is just a wagon train
I'm glad you're not a loser like me"
Losers, losers
Ten gallon bruisers
From Genghis Khan to the Fuller Brush Man
They're just a bunch of losers like me
Catherine had taken Bodie to Jameson Queen Animal Hospital in a taxi. I was following her progress on the Find-My app while watching Johnny Strides walk down Roncesvalles Ave on live TV, knowing that he wasn't far away from her and heading haphazardly in her direction.
She had already gone into the taxi for her return trip home when Johnny was still at Sorauren Ave. I hadn't realized it on the live stream but checking the You-Tube video, he managed to capture the taxi driving by! Catherine and Bodie are in the back seat:
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I found a loop in Éric Angelini's "confined" sequence (about which I wrote last month). Term #60614674264 (= 27651356989742597468495745) is a duplicate of term #18563532230. Differences in the lead-up terms are highlighted here:
#18563532226 6912789247435649367123936 #60614674260 6912789247185649367123936
#18563532227 13825578494871298734247872 #60614674261 13825578494371298734247872
#18563532228 13825678494871298734247872 #60614674262 13825678494371298734247872
#18563532229 27651356989742597468495744 #60614674263 27651356988742597468495744
#18563532230 27651356989742597468495745 = #60614674264 27651356989742597468495745
So we have a loop of length 42051142034. The smallest term in the loop appears to be 507434154592, so here is an abridged loop sequence (asterisk denotes the largest term; three twelve-digit local minima are also shown; indices of all these corrected February 29):
0 507434154592
1 1014868309184
2 2029736618368
3 2029736718368
4 4059473436736
5 8118946873472
6 8128946873472
7 16257893746944
8 16257893746945
9 32515787493890
10 65031574987780
11 65031574987880
12 65031574987890
13 130063149975780
14 131631410075780
15 13163141175780
16 13163141275780
17 26326282551560
18 26326282561560
19 52652565123120
20 105305130246240
... ...
17074586421 49512395802029907136051366345193519491458782692496790312698501120
17074586422 495123958020210007136051367345193519491458782692496790312698501220 *
17074586423 4951239580202117136051367345193519491458782692496790312698501230
... ...
25756695203 5007793970328
25756695204 517893970328
25756695205 1035787940656
... ...
25757984145 5097006463136
25757984146 509716463136
25757984147 1019432926272
... ...
27813217917 6806950060736
27813217918 680695160736
27813217919 1361390321472
... ...
42051142014 1128050902650182
42051142015 1228050902650182
42051142016 1238050902650182
42051142017 2476101805300364
42051142018 247610180531364
42051142019 495220361062728
42051142020 495230361062728
42051142021 990460722125456
42051142022 1000460723125456
42051142023 11460723125456
42051142024 12460723125456
42051142025 24921446250912
42051142026 24921456250912
42051142027 49842912501824
42051142028 99685825003648
42051142029 10068582513648
42051142030 1168582513648
42051142031 1268582513648
42051142032 2537165027296
42051142033 5074330054592
42051142034 507434154592
In Éric Angelini's latest effort, he posits some interesting sequences. Specifically, half-way down the page, we have "replace the chunk by [the chunk + 1]". In case this is not entirely clear, allow me to restate the rule. Any integer that contains one or more blocks of identical adjacent digits evolves into another integer where each of these blocks is replaced with the value of the block plus one. Thus 133555777799999000000 becomes 13455677781000001. The two 3s are replaced with 34, the three 5s with 556, the four 7s with 7778, the five 9s with 100000, and the six 0s with 1. If our integer does not contain any blocks of identical adjacent digits, it becomes twice that integer. A starting integer evolves by the repeated application of these rules:
133555777799999000000
13455677781000001
1345667788111
1345677889112
1345678899122
13456789100123
1345678911123
1345678911223
1345678912233
1345678912334
1345678912344
1345678912345
2691357824690
5382715649380
10765431298760
21530862597520
43061725195040
86123450390080
8612345039180
17224690078360
1723469178360
...
Starting with the integer 1, Giorgos Kalogeropoulos makes the evolution out to be:
1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 65636, 131272, 262544, 262545, 525090, 1050180, 2100360, 211360, 212360, 424720, 849440, 849450, 1698900, 169891, 339782, 349782, 699564, 6100564, 611564, 612564, 1225128, 1235128, 2470256, 4940512, 9881024, 9891024, 19782048, 39564096, 79128192, 158256384, 316512768, 633025536, 634025636, 1268051272, 2536102544, 2536102545, 5072205090, 5072305090, 10144610180, 10145610180, 20291220360, 20291230360, 40582460720, 81164921440, 81264921450, 162529842900, 16252984291, 32505968582, ...
He suggests that the sequence seems to "explode to infinity". Actually, that initial explosion levels off after a thousand or so terms:
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I was sufficiently interested in this sequence to generate 15 billion terms. I graphed only the local minima and maxima, one each for every million terms. The initial explosion terms are ignored by setting the first minimum to 114782627657382. This way we see the sequence's confined space. Three extrema (one maximum, two 11-digit minima) are identified:
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The minima and maxima medians are ~10^18 and ~10^54. Because of the confined space the sequence will evolve into a loop, but particulars about this loop might never be known. To get a sense of this, be aware that in the graphed 15 billion sequence terms there are only 15 confined 11-digit integers. An additional 23 exist at the start but I cannot include these as being confined. So the sequence generates about one 11-digit integer every 10^9 terms. It could of course be more, or less, because the statistical estimate is based empirically on the 15 billion terms that we have so far examined.
How many random 11-digit integers are required in order to have a 50% chance that two of them are duplicates? It is roughly 350000. So, we need to generate some 350*10^3*10^9 = 350 trillion terms in order to have a decent shot at finding a loop.
