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

1,2

Comments

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

Links

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

Example

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.

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]

Prog

(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

Crossrefs

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

Sequence in context: A218524 A188863 A228687 * A223332 A340696 A333428

Adjacent sequences: A349106 A349107 A349108 * A349110 A349111 A349112

Keyword

nonn

Author

Amiram Eldar, Nov 08 2021

Status

approved