A348583 Numbers k such that k | A002129(k).

1, 60, 728, 6960, 60512, 97152, 728000, 1900080, 2184000, 4371840, 26522496, 843480000, 23009688000, 46352390400, 93155148800, 279465446400, 701869363200, 938948846080, 1099176108032, 2816846538240

Offset: 1

Author

Amiram Eldar, Oct 24 2021

Keywords

• nonn
• more

Comments

Equivalently, numbers k such that k | A113184(k).

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, ...

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.

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.

Examples

			60 is a term since A002129(60) = -120 is divisible by 60.
		

Links

• Index entries for sequences where any odd perfect numbers must occur

Crossrefs

Cf. A000203, A002129, A005382, A113184, A167917.

Programs

Mathematica

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[#], #] &]

Extensions

a(20) from Martin Ehrenstein, Nov 06 2021