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

1,2

Links

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

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.

Mathematica

f[p_, e_] := p^e - If[p == 2, 0, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

Prog

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

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).

Sequence in context: A108514 A317419 A384252 * A364843 A372678 A120456

Adjacent sequences: A384053 A384054 A384055 * A384057 A384058 A384059

Keyword

nonn,easy,mult

Author

Amiram Eldar, May 18 2025

Status

approved