%I #22 Oct 31 2020 01:39:59

%S 1,3,7,15,31,21,127,945,4095,1023,544635,46035,8191,355725,279279,

%T 1796165,5388495,180213,131071,110800305,249987465,42833475,3112865,

%U 8109585,524287,9338595,49198149,253255275,22017975903,13341660255,30101431815,709933138551

%N The odd part of those numbers which divide the sum of their divisors (A007691).

%C The data was derived from the data in A007691.

%C The Mersenne primes A000668 are a subsequence; in fact a prime is in this sequence if and only if it is a Mersenne prime.

%C Note that John Voight's 'spoof odd perfect number' 22017975903 is included. - _Peter Luschny_, Oct 30 2020

%H Alois P. Heinz, <a href="/A320024/b320024.txt">Table of n, a(n) for n = 1..1600</a>

%H BYU Computational Number Theory Group, <a href="https://arxiv.org/abs/2006.10697">Odd, spoof perfect factorizations</a>, arXiv:2006.10697 [math.NT], 2020.

%H J. Voight, <a href="https://math.dartmouth.edu/~jvoight/articles/opn-mass-rev-060211.pdf">On the nonexistence of odd perfect numbers</a>.

%F a(n) = A000265(A007691(n)).

%p a_list := proc(len) local L, n; L := NULL; for n from 1 to len do

%p if numtheory:-sigma(n) mod n = 0 then L := L, n/2^padic[ordp](n, 2) fi od; L end:

%t A007691 = Select[Range[1000000], Divisible[DivisorSigma[1, #], #] &]; Table[A007691[[n]]/2^IntegerExponent[A007691[[n]], 2], {n, 1, Length[A007691]}] (* _Vaclav Kotesovec_, Oct 14 2018 *)

%Y Cf. A000265, A000668, A007691.

%K nonn

%O 1,2

%A _Peter Luschny_, Oct 03 2018