A360541 a(n) is the least number k such that k*n is a cubefull number (A036966).

1, 4, 9, 2, 25, 36, 49, 1, 3, 100, 121, 18, 169, 196, 225, 1, 289, 12, 361, 50, 441, 484, 529, 9, 5, 676, 1, 98, 841, 900, 961, 1, 1089, 1156, 1225, 6, 1369, 1444, 1521, 25, 1681, 1764, 1849, 242, 75, 2116, 2209, 9, 7, 20, 2601, 338, 2809, 4, 3025, 49, 3249, 3364

Offset: 1

Author

Amiram Eldar, Feb 11 2023

Keywords

• nonn
• easy
• mult

Links

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

Crossrefs

Cf. A036966, A329376, A356193, A360539.

Programs

Mathematica

f[p_, e_] := p^(Max[e, 3] - e); a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

PARI

a(n) = {my(f = factor(n)); prod(i=1, #f~, f[i, 1]^(max(f[i, 2], 3) - f[i, 2]));}

Formula

a(n) = 1 if and only if n is cubefull number (A036966).

a(n) = A356193(n)/n.

a(n) = A360539(n)^2/A329376(n)^3.

Multiplicative with a(p^e) = p^(max(e, 3) - e), i.e., a(p) = p^2, a(p^2) = p, and a(p^e) = 1 for e >= 3.

Dirichlet g.f.: zeta(s) * Product_{p prime} (1 + p^(2-s) - p^(-s) - p^(2-2*s) + p^(1-2*s) - p^(1-3*s) + p^(-3*s)).

Sum_{k=1..n} a(k) ~ c * n^3, where c = (zeta(3)/3) * Product_{p prime} (1 - 1/p^2 - 1/p^3 + 2/p^5 - 1/p^6 - 1/p^8 + 2/p^9 - 1/p^10) = 0.2078815423... .