cp's OEIS Frontend

A384786 Numbers with a record number of unordered factorizations into 1 and prime powers p^e where p is prime and e >= 3 (A246549).

%I A384786 #10 Jun 10 2025 11:36:37
%S A384786 1,64,256,512,1024,2048,4096,8192,16384,32768,65536,131072,262144,
%T A384786 524288,1048576,2097152,4194304,8388608,16777216,33554432,67108864,
%U A384786 134217728,268435456,536870912,1073741824,2147483648,4294967296,8589934592,17179869184,34359738368,68719476736
%N A384786 Numbers with a record number of unordered factorizations into 1 and prime powers p^e where p is prime and e >= 3 (A246549).
%C A384786 The least term that is not a power of 2 is a(85) = 2^61 * 3^18.
%C A384786 Indices of records of the multiplicative function f(n) with f(p^e) = A008483(e).
%C A384786 All the terms are cubefull numbers since f(1) = 1 and f(n) = 0 if n is a noncubefull number.
%C A384786 The corresponding record values are 1, 2, 3, 4, 5, 6, 9, 10, 13, 17, 21, 25, 33, 39, 49, ... (see the link for more values).
%H A384786 Amiram Eldar, <a href="/A384786/b384786.txt">Table of n, a(n) for n = 1..567</a>
%H A384786 Amiram Eldar, <a href="/A384786/a384786.txt">Table of n, a(n), and record values for n = 1..567</a>
%H A384786 <a href="/index/Pow#powerful">Index entries for sequences related to powerful numbers</a>.
%t A384786 f[p_, e_] := PartitionsP[e] - PartitionsP[e-1] - PartitionsP[e-2] + PartitionsP[e-3]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; With[{lps = Cases[Import["https://oeis.org/A025487/b025487.txt", "Table"], {_, _}][[;; , 2]]}, sm = -1; seq = {}; Do[s1 = s[lps[[i]]]; If[s1 > sm, sm = s1; AppendTo[seq, lps[[i]]]], {i, 1, Length[lps]}]; seq]
%Y A384786 Subsequence of A001694 and A025487 (i.e., of A181800).
%Y A384786 Cf. A008483, A046055, A246549, A384784 (powerful analog), A384785.
%K A384786 nonn
%O A384786 1,2
%A A384786 _Amiram Eldar_, Jun 10 2025