A384784 Numbers with a record number of unordered factorizations into 1 and prime powers p^e where p is prime and e >= 2 (A025475).

1, 16, 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

Offset

1,2

Comments

The least term that is not a power of 2 is a(47) = 2^35 * 3^10.

Indices of records in A188585.

All the terms are powerful numbers since A188585(1) = 1 and A188585(n) = 0 if n is a nonpowerful number.

The corresponding record values are 1, 2, 4, 7, 8, 12, 14, 21, 24, 34, 41, 55, 66, 88, ... (see the link for more values).

Links

Amiram Eldar, Table of n, a(n) for n = 1..713

Amiram Eldar, Table of n, a(n), A188585(a(n)) for n = 1..713

Index entries for sequences related to powerful numbers.

Mathematica

f[p_, e_] := PartitionsP[e] - PartitionsP[e-1]; 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. A025475, A046055, A188585, A384783, A384786 (cubefull analog).

Sequence in context: A317235 A375573 A255661 * A073718 A230970 A352738

Adjacent sequences: A384781 A384782 A384783 * A384785 A384786 A384787

Keyword

nonn

Author

Amiram Eldar, Jun 10 2025

Status

approved