A057523 -id:A057523 - cp's OEIS frontend

A370256 The number of ways in which n can be expressed as b^2 * c^3, with b and c >= 1.

1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0

Offset: 1

Amiram Eldar, Feb 23 2024

First differs from A075802 and A112526 at n = 64.
The least number k such that a(k) = n is A005179(n)^6.
The indices of records are the sixth powers of the highly composite numbers, A002182(n)^6.

			1 = 1^2 * 1^3, so a(1) = 1.
64 = 1^2 * 4^3 = 8^2 * 1^3, so a(64) = 2.
4096 = 64^2 * 1^3 = 8^2 * 4^3 = 1^2 * 16^3, so a(4096)= 3.

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

Cf. A000005, A001694, A002182, A005179, A057523, A075802, A078434, A103221, A112526.

f[p_, e_] := Floor[(e + 2)/2] - Floor[(e + 2)/3]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = vecprod(apply(x -> (x+2)\2 - (x+2)\3, factor(n)[, 2]));

for(n=1, 100, print1(direuler(p=2, n, 1/((1 - X^2)*(1 - X^3)))[n], ", ")) \\ Vaclav Kotesovec, Feb 23 2024

from math import prod
from sympy import factorint
def A370256(n): return prod((e>>1)+1-(e+2)//3 for e in factorint(n).values()) # Chai Wah Wu, Apr 15 2025

Multiplicative with a(p^e) = A103221(e).
a(n) > 0 if and only if n is a powerful number (A001694).
a(A001694(n)) = A057523(n).
a(n^6) = A000005(n).
Sum_{k=1..n} a(k) ~ zeta(3/2) * sqrt(n) + zeta(2/3) * n^(1/3).
Dirichlet generating function: zeta(2*s)*zeta(3*s). - Vaclav Kotesovec, Feb 23 2024

A384789 The number of ways in which the n-th cubefull number can be expressed as b^3 * c^4 * d^5, with b, c and d >= 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 4, 1, 1

Offset: 1

Amiram Eldar, Jun 10 2025

The positive values of the multiplicative function f(n) with f(p^e) = A008680(e). Or, equivalently, a(n) is the value of this function at A036966(n).

			a(12) = 2 since A036966(12) = 256 = 2^8 has 2 representations as b^3*c^4*d^5: 2^3 * 2^5 (b = d = 2, c = 1) and 4^4 (b = d = 1, c = 4).
a(38) = 3 since A036966(38) = 4096 = 2^12 has 3 representations as b^3*c^4*d^5: 2^3 * 2^4 * 2^5 (b = c = d = 2), 8^4 (b = d = 1, c = 8) and 16^3 (b = 16, c = d = 1).

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

Cf. A008680, A036966, A057523 (powerful analog), A384791.

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]; Select[Array[s, 10000], # > 0 &]

f(e) = floor((1+(-1)^e)*(-1)^floor(e/2)/8 + (e^2 + 12*e + 90)/120);
list(kmax) = {my(e); for(k = 1, kmax, e = factor(k)[, 2]; if(k == 1 || vecmin(e) > 2, print1(vecprod(apply(x -> f(x), e)), ", ")));}

a(n) >= 1.

A384790 Numbers with a record number of ways in which they can be expressed as b^2 * c^3, with b and c >= 1.

Original entry on oeis.org

1, 64, 4096, 46656, 2985984, 191102976, 2176782336, 12230590464, 46656000000, 2985984000000, 34012224000000, 191102976000000, 2176782336000000, 139314069504000000, 351298031616000000, 4001504141376000000, 22483074023424000000, 256096265048064000000, 16390160963076096000000

Offset: 1

Amiram Eldar, Jun 10 2025

nonn

Indices of records of A370256.
All the terms are powerful numbers since A370256(1) = 1 and A370256(n) = 0 if n is a nonpowerful number.
The corresponding record values are 1, 2, 3, 4, 6, 8, 9, 10, 12, 16, 18, 20, ... (see the link for more values).

Amiram Eldar, Table of n, a(n) for n = 1..61.
Amiram Eldar, n, a(n), A370256(a(n)) for n = 1..61.
Index entries for sequences related to powerful numbers.

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

Cf. A002182 (sixth root), A046055, A057523, A370256, A384791 (cubefull analog).

f[p_, e_] := Floor[(e + 2)/2] - Floor[(e + 2)/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]

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A370256 The number of ways in which n can be expressed as b^2 * c^3, with b and c >= 1.

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A384789 The number of ways in which the n-th cubefull number can be expressed as b^3 * c^4 * d^5, with b, c and d >= 1.

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A384790 Numbers with a record number of ways in which they can be expressed as b^2 * c^3, with b and c >= 1.

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Author

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Crossrefs

Programs

Mathematica