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%I #7 Nov 08 2021 04:29:05
%S 128,729,900,4900,10404,17424,24336,52900,78400,79524,81796,297025,
%T 304175,304200,313600,346921,417316,532900,1612900,1656200,1960000,
%U 2238016,2464900,3129361,3232804,3334276,3496900,3534400,3992004,6056521,6974881,9245000,10672200
%N Powerful numbers (A001694) whose sum of aliquot powerful divisors (including 1) is larger than 1 and is also powerful.
%C Numbers k such that A112526(k) = A112526(A183097(k) - k) = 1.
%e 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.
%t 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]
%o (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
%Y Cf. A001694, A183097, A349109.
%K nonn
%O 1,1
%A _Amiram Eldar_, Nov 08 2021
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