|
%I #9 Feb 13 2020 20:16:22
%S 153,325,369,657,725,801,833,845,873,925,1017,1233,1325,1377,1445,
%T 1525,1737,2009,2057,2097,2169,2313,2525,2529,2725,2817,2925,3033,
%U 3177,3321,3577,3609,3681,3725,3757,3897,3925,4041,4113,4205,4325,4361,4525,4689,4753,4901,4925,4961,5121,5193,5337,5409,5537,5553,5725
%N Odd numbers n, not powers of primes, such that sigma(n) is congruent to 2 modulo 8.
%C Proof that any odd perfect number, if such numbers exist at all, has to reside in this sequence: As all terms in A228058 are = 1 modulo 4 (their binary expansion ends as "01"), and taking sigma of an odd perfect number would multiply it by two (shift one bit-position left), the base-2 expansion of that result would end as "010"), i.e., sigma(k) modulo 8 should be 2 (not 6) for such numbers k.
%H Antti Karttunen, <a href="/A332228/b332228.txt">Table of n, a(n) for n = 1..25000</a>
%H <a href="/index/O#opnseqs">Index entries for sequences where any odd perfect numbers must occur</a>
%H <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%o (PARI) isA332228(n) = ((n%2)&&!isprimepower(n)&&2==(sigma(n)%8));
%Y Cf. A000203, A332229.
%Y Subsequence of A228058, of A332226 and of A332227.
%K nonn
%O 1,1
%A _Antti Karttunen_, Feb 13 2020
|
