A384053 The number of integers k from 1 to n such that the greatest divisor of k that is a unitary divisor of n is a cube.

1, 1, 2, 3, 4, 2, 6, 8, 8, 4, 10, 6, 12, 6, 8, 15, 16, 8, 18, 12, 12, 10, 22, 16, 24, 12, 27, 18, 28, 8, 30, 31, 20, 16, 24, 24, 36, 18, 24, 32, 40, 12, 42, 30, 32, 22, 46, 30, 48, 24, 32, 36, 52, 27, 40, 48, 36, 28, 58, 24, 60, 30, 48, 64, 48, 20, 66, 48, 44

Offset: 1

Author

Amiram Eldar, May 18 2025

Keywords

• nonn
• easy
• mult

Links

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

Crossrefs

Unitary analog of A078429.

Cf. A013664, A384046, A384047.

The number of integers k from 1 to n such that the greatest divisor of k that is a unitary divisor of n is: A047994 (1), A384048 (squarefree), A384049 (cubefree), A384050 (powerful), A384051 (cubefull), A384052 (square), this sequence (cube), A384054 (exponentially odd), A384055 (odd), A384056 (power of 2), A384057 (3-smooth), A384058 (5-rough).

Programs

Mathematica

f[p_, e_] := p^e - If[Divisible[e, 3], 0, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a,100]

PARI

a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^f[i,2] - if(f[i,2]%3, 1, 0));}

Formula

Multiplicative with a(p^e) = p^e if e is a multiple of 3, and p^e-1 otherwise.

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

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = zeta(6) * Product_{p prime} (1 - 1/p^2 + 1/p^3 - 1/p^4 + 1/p^5 - 1/p^6) = 0.71190515701599590826... .