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

1, 64, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33554432, 67108864, 134217728, 268435456, 536870912, 1073741824, 2147483648, 4294967296, 8589934592, 17179869184, 34359738368, 68719476736

Offset: 1

Author

Amiram Eldar, Jun 10 2025

Keywords

• nonn

Comments

The least term that is not a power of 2 is a(85) = 2^61 * 3^18.

Indices of records of the multiplicative function f(n) with f(p^e) = A008483(e).

All the terms are cubefull numbers since f(1) = 1 and f(n) = 0 if n is a noncubefull number.

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).

Links

• Amiram Eldar, Table of n, a(n) for n = 1..567
• Amiram Eldar, Table of n, a(n), and record values for n = 1..567
• Index entries for sequences related to powerful numbers.

Crossrefs

Subsequence of A001694 and A025487 (i.e., of A181800).

Cf. A008483, A046055, A246549, A384784 (powerful analog), A384785.

Programs

Mathematica

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]