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 A349109 Powerful numbers (A001694) whose sum of powerful divisors (including 1) is also powerful. 1
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Numbers k such that A112526(k) = A112526(A183097(k)) = 1. LINKS 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 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

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Last modified January 22 00:08 EST 2022. Contains 350481 sequences. (Running on oeis4.)