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

1,2

Links

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

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... .

Mathematica

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

Prog

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

Crossrefs

Cf. A036966, A329376, A356193, A360539.

Sequence in context: A253288 A358971 A358786 * A365298 A367932 A048798

Adjacent sequences: A360538 A360539 A360540 * A360542 A360543 A360544

Keyword

nonn,easy,mult

Author

Amiram Eldar, Feb 11 2023

Status

approved