A349110 Powerful numbers (A001694) whose sum of aliquot powerful divisors (including 1) is larger than 1 and is also powerful.

128, 729, 900, 4900, 10404, 17424, 24336, 52900, 78400, 79524, 81796, 297025, 304175, 304200, 313600, 346921, 417316, 532900, 1612900, 1656200, 1960000, 2238016, 2464900, 3129361, 3232804, 3334276, 3496900, 3534400, 3992004, 6056521, 6974881, 9245000, 10672200

Offset: 1

Author

Amiram Eldar, Nov 08 2021

Keywords

• nonn

Comments

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

Examples

			128 = 2^7 is a term since it is powerful and the sum of its aliquot powerful divisors, A183097(128) - 128 =  1 + 4 + 8 + 16 + 32 + 64 = 125 = 5^3 is also powerful.
		

Crossrefs

Cf. A001694, A183097, A349109.

Programs

Mathematica

powQ[n_] := 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] - n]; Select[Range[1.1*10^7], q]

PARI

isok(n) = my(s); ispowerful(n) && (s=sumdiv(n, d, if (d1) && ispowerful(s); \\ Michel Marcus, Nov 08 2021