A384055 The number of integers k from 1 to n such that the greatest divisor of k that is a unitary divisor of n is odd.

1, 1, 3, 3, 5, 3, 7, 7, 9, 5, 11, 9, 13, 7, 15, 15, 17, 9, 19, 15, 21, 11, 23, 21, 25, 13, 27, 21, 29, 15, 31, 31, 33, 17, 35, 27, 37, 19, 39, 35, 41, 21, 43, 33, 45, 23, 47, 45, 49, 25, 51, 39, 53, 27, 55, 49, 57, 29, 59, 45, 61, 31, 63, 63, 65, 33, 67, 51, 69

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 A026741.

Cf. A000265, A006519, 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), A384053 (cube), A384054 (exponentially odd), this sequence (odd), A384056 (power of 2), A384057 (3-smooth), A384058 (5-rough).

Programs

Mathematica

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

Formula

Multiplicative with a(2^e) = 2^e-1, and p^e if p is an odd prime.

a(n) = n * A047994(n) / A384056(n).

a(n) = A047994(A006519(n)) * A000265(n).

Dirichlet g.f.: zeta(s-1) * (1 - 1/2^(s-1) + 1/2^(2*s-1))/(1 - 1/2^s).

Sum_{k=1..n} a(k) ~ (5/12) * n^2.