A384791 Numbers with a record number of ways in which they can be expressed as b^3 * c^4 * d^5, with b, c and d >= 1.

1, 256, 4096, 32768, 262144, 1048576, 8388608, 16777216, 134217728, 268435456, 1073741824, 4294967296, 8589934592, 34359738368, 68719476736, 110075314176, 549755813888, 557256278016, 1761205026816, 4458050224128, 7044820107264, 8916100448256, 56358560858112, 71328803586048

Offset

1,2

Comments

The least term that is not a power of 2 is a(16) = 2^24 * 3^8.

Indices of records of the multiplicative function f(n) with f(p^e) = A008680(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, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, ... (see the link for more values).

Every exponent must be the index of the first occurrence of A008680(e) in A008680. So possible exponents of prime factors of terms are 0, 8, 12, 15, 18, 20, ... - David A. Corneth, Jun 30 2025

Links

David A. Corneth, Table of n, a(n) for n = 1..810 (first 216 terms from Amiram Eldar, terms <= 10^120)

David A. Corneth, Table of n, a(n), exponents of prime factors and record values for n = 1..810

Index entries for sequences related to powerful numbers.

Example

256 in the sequence as 256 = 1^3 * 4^4 * 1^5 = 2^3 * 1^4 * 2^5 so there are two ways to write 256 as b^3 * c^4 * d^5, with b, c and d >= 1 and no smaller positive integer can be written in at least two ways like that. - David A. Corneth, Jun 30 2025

Mathematica

f[p_, e_] := Floor[(1+(-1)^e)*(-1)^Floor[e/2]/8 + (e^2 + 12*e + 90)/120]; 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 A025487, A036966 and A181800.

Cf. A008680, A046055, A384789, A384790 (powerful analog).

Sequence in context: A186554 A200843 A016804 * A115111 A200790 A206110

Adjacent sequences: A384788 A384789 A384790 * A384792 A384793 A384794

Keyword

nonn

Author

Amiram Eldar, Jun 10 2025

Status

approved