Problems & Puzzles: Puzzles

Puzzle 670 The same sum of divisors

Shyam Sunder Gupta asks  in his CYF NO. 29  for three consecutive numbers whose sum of divisors is same.

Mukesh Kr Jha dt comments at 23-07-2012: No solution found for this problem for n from 1 to 3,00,00,000 [sic] using Maple application.

J. K. Andersen comments at 28-12-12: http://oeis.org/A002961 shows cases of two consecutive numbers. The table by T. D. Noe of terms up to 10^12 has no cases of three consecutive numbers.

Carlos Rivera has counted the 4804 cases listed by T. D. Noe, grouped by its last digit hoping to find a clue to speed up this search.

Last digit Frequency
0 71
1 369
2 383
3 395
4 1206
5 1229
6 352
7 357
8 374
9 68
 Total 4804

 Q. Send the results of your search.

 


Contributions came from Jan van Delden

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Jan wrote:

Factoring numbers of this size requires a considerable time.
I constructed a large sieve to this purpose, but that doesn’t seem to help a lot.
The best I can do (without other tricks) is extend the given list by a few numbers:
 
Number: 1000001331209 #Divisors: 4 Divisors: 19 1999 3511 7499 sum: 13028
Number: 1000001331210 #Divisors: 7 Divisors: 2 3 5 7 127 4441 8443 sum: 13028
Number: 1000006403131 #Divisors: 3 Divisors: 113 15467 572161 sum: 587741
Number: 1000006403132 #Divisors: 5 Divisors: 2 2 11 41617 546109 sum: 587741
Number: 1000009265660 #Divisors: 6 Divisors: 2 2 5 251 6827 29179 sum: 36266
Number: 1000009265661 #Divisors: 6 Divisors: 3 3 7 271 1709 34273 sum: 36266
Number: 1000010921006 #Divisors: 5 Divisors: 2 107 281 1913 8693 sum: 10996
Number: 1000010921007 #Divisors: 6 Divisors: 3 7 7 239 4759 5981 sum: 10996
Number: 1000011047139 #Divisors: 7 Divisors: 3 3 13 13 67 211 46507 sum: 46817
Number: 1000011047140 #Divisors: 7 Divisors: 2 2 5 7 23 8011 38767 sum: 46817
Number: 1000018797129 #Divisors: 7 Divisors: 3 3 3 11 41 4517 18181 sum: 22759
Number: 1000018797130 #Divisors: 6 Divisors: 2 5 19 181 1373 21179 sum: 22759
Number: 1000019050750 #Divisors: 7 Divisors: 2 5 5 5 97 1861 22159 sum: 24134
Number: 1000019050751 #Divisors: 4 Divisors: 7 1117 9391 13619 sum: 24134
Number: 1000021212575 #Divisors: 8 Divisors: 5 5 7 7 11 181 263 1559 sum: 2038
Number: 1000021212576 #Divisors: 13 Divisors: 2 2 2 2 2 3 3 3 3 3 199 487 1327 sum: 2038
Number: 1000022168025 #Divisors: 7 Divisors: 3 3 3 5 5 3847 385109 sum: 388975
Number: 1000022168026 #Divisors: 6 Divisors: 2 7 7 41 641 388277 sum: 388975
Number: 1000022709374 #Divisors: 6 Divisors: 2 13 83 127 1427 2557 sum: 4209
Number: 1000022709375 #Divisors: 11 Divisors: 3 3 3 3 5 5 5 5 5 1453 2719 sum: 4209
Number: 1000023240048 #Divisors: 10 Divisors: 2 2 2 2 3 7 7 229 587 3163 sum: 4004
Number: 1000023240049 #Divisors: 4 Divisors: 929 1013 1013 1049 sum: 4004
Number: 1000023850088 #Divisors: 8 Divisors: 2 2 2 37 61 127 131 3329 sum: 3691
Number: 1000023850089 #Divisors: 7 Divisors: 3 3 11 71 167 269 3167 sum: 3691
Number: 1000025351679 #Divisors: 6 Divisors: 3 17 83 271 499 1747 sum: 2620
Number: 1000025351680 #Divisors: 13 Divisors: 2 2 2 2 2 2 2 2 2 5 347 751 1499 sum: 2620
Number: 1000028446831 #Divisors: 4 Divisors: 19 223 11867 19889 sum: 31998
Number: 1000028446832 #Divisors: 8 Divisors: 2 2 2 2 83 113 211 31583 sum: 31998
Number: 1000030941450 #Divisors: 9 Divisors: 2 3 3 5 5 23 101 163 5869 sum: 6174
Number: 1000030941451 #Divisors: 4 Divisors: 137 1499 1741 2797 sum: 6174
Number: 1000033100857 #Divisors: 6 Divisors: 7 7 7 17 8623 19889 sum: 28550
Number: 1000033100858 #Divisors: 5 Divisors: 2 11 599 3049 24889 sum: 28550
Number: 1000036424168 #Divisors: 6 Divisors: 2 2 2 37 41621 81173 sum: 122837
Number: 1000036424169 #Divisors: 5 Divisors: 3 3 47 23911 98873 sum: 122837
Number: 1000039050464 #Divisors: 9 Divisors: 2 2 2 2 2 127 359 433 1583 sum: 2512
Number: 1000039050465 #Divisors: 8 Divisors: 3 5 7 13 19 37 557 1871 sum: 2512
Number: 1000045099151 #Divisors: 6 Divisors: 7 17 37 307 601 1231 sum: 2200
Number: 1000045099152 #Divisors: 12 Divisors: 2 2 2 2 3 3 3 3 11 67 811 1291 sum: 2200
Number: 1000056935865 #Divisors: 7 Divisors: 3 5 13 89 151 263 1451 sum: 1975
Number: 1000056935866 #Divisors: 6 Divisors: 2 23 109 379 641 821 sum: 1975
Number: 1000061081071 #Divisors: 5 Divisors: 13 157 433 761 1487 sum: 2851
Number: 1000061081072 #Divisors: 9 Divisors: 2 2 2 2 17 61 73 353 2339 sum: 2851

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