# [seqfan] Duplicate sigma values

franktaw at netscape.net franktaw at netscape.net
Fri Jun 19 02:52:05 CEST 2009

```Recent discussion got me thinking about numbers with the same values
for two or more sigma_k(n).  My first thought was to look at duplicates
in sigma and sigma_2; i.e., sigma(n) = sigma(m) and sigma_2(n) =
sigma_2(m).  However, I decided to look first at the simpler case of
duplicates in sigma_0 (A000005) and sigma.

For values of sigma up to 1000, there are the following duplications:

4 24: 14,15
4 48: 33,35
4 72: 46,51,55
4 96: 62,69
4 120: 87,95
8 120: 54,56
4 144: 94,115,119
8 144: 66,70
4 168: 123,143
4 180: 118,145
4 192: 141,155,161
4 216: 142,159,187
8 216: 102,110
4 240: 158,177,203,209
8 240: 114,135
4 252: 166,205,221
4 288: 213,235,253
8 288: 138,154,165
4 324: 214,265
4 336: 249,287,299
8 336: 182,195
12 336: 132,140
4 360: 267,295,319,323
8 360: 174,184,190
4 384: 254,329,341
8 384: 186,231
4 408: 303,335
4 420: 278,377
4 432: 321,355,371,391
8 432: 230,238,255
6 434: 244,325
4 456: 302,339,407
4 480: 395,413,437
8 480: 248,266,285,297
4 504: 334,415,451
8 504: 246,286
12 504: 104,220,224
4 528: 393,473
4 540: 358,445,493
4 576: 382,497,517,527
8 576: 282,310,322,345,357,385
4 588: 485,533
4 600: 398,447,551
4 640: 553,589
4 648: 535,583
8 648: 318,374
4 672: 446,501,581,611
8 672: 429,455
12 672: 276,308
4 684: 454,565,629
4 720: 478,537,623,649,667
8 720: 354,376,406,418,435,459
16 720: 264,270,280
4 756: 502,689,697
8 756: 410,442
12 756: 340,352
4 768: 573,635,713
8 768: 434,465,483
4 792: 526,591,655,731
6 798: 452,605,637
4 816: 542,707,737
4 840: 695,767,779
12 840: 348,380
4 864: 749,781,799
8 864: 426,470,506,561,595
4 880: 763,817
4 912: 681,755,851
8 912: 518,555
4 936: 622,699
12 936: 414,495
4 960: 717,869,893,899
8 960: 474,609,621,627,665

In these, the first two are the values of sigma_0 and sigma; the rest
is the list of numbers duplicating those values.

Some questions: are there any such values with sigma_0 odd?
(Equivalently, where n and m are squares.)  More generally, what values
for sigma_0 can occur?  It is certain that the possible values are
closed under multiplication by arbitrary integers, since if n and m
have the same values for sigma_0 and sigma, and a is any number
relatively prime to both n and m, then an and am also have the same
values for sigma_0 and sigma, and in particular sigma_0(an) =
sigma_0(a)*sigma_0(n) -- and we can take sigma_0(a) to be an arbitrary
positive integer.  On the other hand, no prime p can occur for
sigma_0(n), since then we would have to have n = q^{p-1} for some prime
q, and any two choices for q would give different values for sigma(n)
and sigma(m).

There seems to be a distinct tendency for numbers with the same values
for sigma to have different sigma_0's.  An extreme case is 744; there
are 7 numbers with sigma(n) = 744; all 7 have distinct values for
sigma_0.

Franklin T. Adams-Watters

P.s. any thoughts on how to present this data in the OEIS?

```

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