Ed Pegg's product partition challenge

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. After first verifying that it is the smallest, I reverse-engineered three somewhat larger products:

  1476395008 = 2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*2*22
116508327936 = 4*4*4*4*4*4*4*4*4*444444
505627938816 = 4*4*4*4*4444*444444
640532803911 = 7*7*7*7*7*7*7*777777

That would be sufficient to create an OEIS sequence, but I need to make sure that there are no additional (in-between) solutions. That will require (I'm guessing) several months!

Pandigital products

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.

A Falcoz digit-fancy

click to enlarge

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.

Stymied by the beast

click to enlarge

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:

click to enlarge

Drive-by

Nearby Denison Rd W was blocked off from the curve at the hidden parkette (first picture, helicopter view) to Clouston Ave (third picture, helicopter view), because of its relevance to a "700 metres away" (Weston Road & Sidney Belsey Crescent) drive-by shooting. There was no explanation given as to why the two locations are thus dissociated but it is reasonable to suppose that the shooting actually happened on Denison Rd W and that the victim managed to make his way to the other location. The second picture is my photograph of a police car blocking access on Lippincott St W.

Barring the existence of folk in the neighbourhood who actually heard/witnessed the drive-by shooting, I'll venture that the location may have been fabricated by the victim.

Seventy

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

Losers

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

Almost

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:

click to enlarge

Confined (a loop)

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
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Back from the vet

Ten-year-old Bodie is back from the veterinarian where, this morning, he had surgery to remove five teeth and a papilloma on his back.