A384050 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 powerful number.

1, 1, 2, 4, 4, 2, 6, 8, 9, 4, 10, 8, 12, 6, 8, 16, 16, 9, 18, 16, 12, 10, 22, 16, 25, 12, 27, 24, 28, 8, 30, 32, 20, 16, 24, 36, 36, 18, 24, 32, 40, 12, 42, 40, 36, 22, 46, 32, 49, 25, 32, 48, 52, 27, 40, 48, 36, 28, 58, 32, 60, 30, 54, 64, 48, 20, 66, 64, 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 A384039.

Cf. A000010, A001694, A055231, A057521, A330596, 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), this sequence (powerful), A384051 (cubefull), A384052 (square), A384053 (cube), A384054 (exponentially odd), A384055 (odd), A384056 (power of 2), A384057 (3-smooth), A384058 (5-rough).

Programs

Mathematica

f[p_, e_] := p^e - If[e < 2, 1, 0]; 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] == 1, 1, 0));}

Formula

Multiplicative with a(p) = p-1, and p^e if e >= 2.

a(n) = n * A047994(n) / A384048(n).

a(n) = A047994(A055231(n)) * A057521(n) = A000010(A055231(n)) * A057521(n).

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

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (1 - 1/p^2 + 1/p^3) = 0.748535... (A330596).