A349109 Powerful numbers (A001694) whose sum of powerful divisors (including 1) is also powerful.

1, 64, 243, 441, 1764, 9800, 15552, 28224, 41616, 60516, 82369, 88200, 189728, 226576, 329476, 336200, 648675, 741321, 968256, 1317904, 1428025, 1707552, 1943236, 2039184, 2056356, 2381400, 2446227, 2798929, 2965284, 2986568, 4372281, 5189400, 5271616, 6508832

Offset: 1

Author

Amiram Eldar, Nov 08 2021

Keywords

• nonn

Comments

Numbers k such that A112526(k) = A112526(A183097(k)) = 1.

Examples

			64 = 2^6 is a term since it is powerful and the sum of its powerful divisors, A183097(64) =  1 + 4 + 8 + 16 + 32 + 64 = 125 = 5^3 is also powerful.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..12154 (terms below 10^19)

Crossrefs

Cf. A001694, A112526, A180090, A183097, A337044, A337045, A349110.

Programs

Mathematica

powQ[n_] := n == 1 || AllTrue[FactorInteger[n][[;;,2]], # > 1 &]; f[p_, e_] := (p^(e + 1) - 1)/(p - 1) - p; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; q[n_] := powQ[n] && powQ[s[n]]; Select[Range[7*10^6], q]

PARI

isok(n) = ispowerful(n) && ispowerful(sumdiv(n, d, d*ispowerful(d))); \\ Michel Marcus, Nov 08 2021

PARI

is(k) = {my(f = factor(k)); ispowerful(f) && ispowerful(prod(i = 1, #f~, (f[i,1]^(f[i,2]+1) - 1)/(f[i,1] - 1) - f[i,1]));} \\ Amiram Eldar, Sep 14 2024