A365207 The number of divisors d of n such that gcd(d, n/d) is a power of 2 (A000079).

1, 2, 2, 3, 2, 4, 2, 4, 2, 4, 2, 6, 2, 4, 4, 5, 2, 4, 2, 6, 4, 4, 2, 8, 2, 4, 2, 6, 2, 8, 2, 6, 4, 4, 4, 6, 2, 4, 4, 8, 2, 8, 2, 6, 4, 4, 2, 10, 2, 4, 4, 6, 2, 4, 4, 8, 4, 4, 2, 12, 2, 4, 4, 7, 4, 8, 2, 6, 4, 8, 2, 8, 2, 4, 4, 6, 4, 8, 2, 10, 2, 4, 2, 12, 4, 4

Offset: 1

Author

Amiram Eldar, Aug 26 2023

Keywords

• nonn
• easy
• mult

Comments

The sum of these divisors is A107749(n).

Links

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

Crossrefs

Cf. A000005, A000079, A000265, A007814, A034444, A038838, A042968, A107749, A122132.

Cf. A001620, A013661, A073002.

Programs

Mathematica

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

PARI

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

Formula

Multiplicative with a(2^e) = e+1 and a(p^e) = 2 for an odd prime p.

a(n) <= A000005(n), with equality if and only if n is in A122132 (or equivalently, n is not in A038838).

a(n) >= A034444(n), with equality if and only if n is not divisible by 4 (A042968).

a(n) = A000005(A006519(n)) * A034444(A000265(n)).

a(n) = A034444(n) * (A007814(n)+1)/2^(1 - (n mod 2)).

Dirichlet g.f.: (4^s/(4^s-1)) * zeta(s)^2/zeta(2*s).

Sum_{k==1..n} a(k) ~ (8/Pi^2)*n*(log(n) + 2*gamma - 2*log(2)/3 - 2*zeta'(2)/zeta(2) - 1), where gamma is Euler's constant (A001620).