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%I #12 Nov 06 2021 09:50:05
%S 1,60,728,6960,60512,97152,728000,1900080,2184000,4371840,26522496,
%T 843480000,23009688000,46352390400,93155148800,279465446400,
%U 701869363200,938948846080,1099176108032,2816846538240
%N Numbers k such that k | A002129(k).
%C Equivalently, numbers k such that k | A113184(k).
%C 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 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 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 <a href="/index/O#opnseqs">Index entries for sequences where any odd perfect numbers must occur</a>
%e 60 is a term since A002129(60) = -120 is divisible by 60.
%t 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 Cf. A000203, A002129, A005382, A113184, A167917.
%K nonn,more
%O 1,2
%A _Amiram Eldar_, Oct 24 2021
%E a(20) from _Martin Ehrenstein_, Nov 06 2021
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