A384783 -id:A384783 - cp's OEIS frontend

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

Amiram Eldar, Jun 10 2025

nonn

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).

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.

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

Cf. A025475, A046055, A188585, A384783, A384786 (cubefull analog).

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]

A384785 The number of unordered factorizations of the n-th cubefull number into 1 and prime powers p^e where p is prime and e >= 3 (A246549).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 3, 1, 1, 4, 1, 1, 2, 1, 1, 5, 1, 1, 2, 1, 1, 6, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 9, 1, 1, 2, 1, 2, 3, 1, 3, 1, 2, 10, 1, 1, 1, 2, 1, 1, 2, 1, 4, 1, 2, 2, 2, 13, 1, 1, 2, 1, 1, 4, 1, 3, 1, 2, 2, 1, 1, 1, 5, 1, 1, 1, 1, 2, 3, 17, 2

Offset: 1

Amiram Eldar, Jun 10 2025

The positive values of the multiplicative function f(n) with f(p^e) = A008483(e). Or, equivalently, a(n) is the value of this function at A036966(n).

			a(6) = 2 since the 6th cubefull number, A036966(6) = 64, has 2 factorizations: 2^3 * 2^3 and 2^6.
a(12) = 3 since the 12th cubefull number, A036966(12) = 256, has 3 factorizations: 2^3 * 2^5, 2^4 * 2^4, and 2^8.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A008483, A036966, A246549, A384783 (powerful analog), A384786.

f[p_, e_] := PartitionsP[e] - PartitionsP[e-1] - PartitionsP[e-2] + PartitionsP[e-3]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; seq[lim_] := Module[{cub = Union[Flatten[Table[i^3*j^4*k^5, {k, 1, Surd[lim, 5]}, {j, 1, Surd[lim/k^5, 4]}, {i, 1, Surd[lim/(j^4*k^5), 3]}]]]}, Select[s /@ cub, # > 0 &]]; seq[10^5]

s(n) = vecprod(apply(x -> numbpart(x)-numbpart(x-1)-numbpart(x-2)+numbpart(x-3), factor(n)[, 2]));
cubs(lim) = {my(c = List()); for(k = 1, sqrtnint(lim, 5), for(j = 1, sqrtnint(lim \ k^5, 4), for(i = 1, sqrtnint(lim \ (j^4*k^5), 3), listput(c, i^3*j^4*k^5)))); Set(c); }
list(lim) = {my(c = cubs(lim), v = List(), s1); for(k = 1, #c, s1 = s(c[k]); if(s1 > 0, listput(v, s1))); Vec(v);}

cp's OEIS Frontend

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

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