A384790 -id:A384790 - cp's OEIS frontend

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

Amiram Eldar, Jun 10 2025

nonn

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

			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

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.

Subsequence of A025487, A036966 and A181800.

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

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]

cp's OEIS Frontend

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.

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