A384056 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 power of 2.

1, 2, 2, 4, 4, 4, 6, 8, 8, 8, 10, 8, 12, 12, 8, 16, 16, 16, 18, 16, 12, 20, 22, 16, 24, 24, 26, 24, 28, 16, 30, 32, 20, 32, 24, 32, 36, 36, 24, 32, 40, 24, 42, 40, 32, 44, 46, 32, 48, 48, 32, 48, 52, 52, 40, 48, 36, 56, 58, 32, 60, 60, 48, 64, 48, 40, 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 A062570.

Cf. A000079, A000265, A006519, A065463, 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), A384055 (odd), this sequence (power of 2), A384057 (3-smooth), A384058 (5-rough).

Programs

Mathematica

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

Formula

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

a(n) = n * A047994(n) / A384055(n).

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

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

Sum_{k=1..n} a(k) ~ (3/5) * c * n^2, where c = Product_{p prime} (1 - 1/(p*(p+1))) = A065463.