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

1,2

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.

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]

Crossrefs

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

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

Sequence in context: A223315 A263924 A236331 * A321070 A017066 A141478

Adjacent sequences: A384783 A384784 A384785 * A384787 A384788 A384789

Keyword

nonn

Author

Amiram Eldar, Jun 10 2025

Status

approved