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

1,1

Comments

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

Links

Table of n, a(n) for n=1..33.

Example

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.

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]

Prog

(PARI) isok(n) = my(s); ispowerful(n) && (s=sumdiv(n, d, if (d<n, d*ispowerful(d)))) && (s>1) && ispowerful(s); \\ Michel Marcus, Nov 08 2021

Crossrefs

Cf. A001694, A183097, A349109.

Sequence in context: A197906 A218903 A333584 * A366827 A297463 A297700

Adjacent sequences: A349107 A349108 A349109 * A349111 A349112 A349113

Keyword

nonn

Author

Amiram Eldar, Nov 08 2021

Status

approved