This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A348583 #12 Nov 06 2021 09:50:05 %S A348583 1,60,728,6960,60512,97152,728000,1900080,2184000,4371840,26522496, %T A348583 843480000,23009688000,46352390400,93155148800,279465446400, %U A348583 701869363200,938948846080,1099176108032,2816846538240 %N A348583 Numbers k such that k | A002129(k). %C A348583 Equivalently, numbers k such that k | A113184(k). %C A348583 The corresponding ratios A002129(k)/k are 1, -2, -2, -3, -2, -3, -3, -4, -4, -4, -4, -4, -4, -4, -3, -4, -4, -3, -2, -4, ... %C A348583 If p is a Mersenne exponent (A000043), and the corresponding Mersenne prime (A000668) M_p = 2^p - 1 is in A005382 or A167917, i.e., 2*M_p - 1 is also a prime, then 2^p*(2^p-1)*(2^(p+1)-3) is a term. The corresponding known terms of this form are 60, 728, 60512, 1099176108032 and 288229001763749888. %C A348583 If a term k is odd, then A002129(k) = A000203(k) and thus k is a multiply-perfect number. Therefore, the odd perfect numbers, if they exist, are terms of this sequence. %H A348583 <a href="/index/O#opnseqs">Index entries for sequences where any odd perfect numbers must occur</a> %e A348583 60 is a term since A002129(60) = -120 is divisible by 60. %t A348583 f[p_, e_] := If[p == 2, 2^(e + 1)-3, (p^(e + 1) - 1)/(p - 1)]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[10^5], Divisible[s[#], #] &] %Y A348583 Cf. A000203, A002129, A005382, A113184, A167917. %K A348583 nonn,more %O A348583 1,2 %A A348583 _Amiram Eldar_, Oct 24 2021 %E A348583 a(20) from _Martin Ehrenstein_, Nov 06 2021