A365207 - OEIS

%I #9 Aug 27 2023 02:00:39

%S 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,

%T 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,

%U 6,4,8,2,8,2,4,4,6,4,8,2,10,2,4,2,12,4,4

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

%C The sum of these divisors is A107749(n).

%H Amiram Eldar, <a href="/A365207/b365207.txt">Table of n, a(n) for n = 1..10000</a>

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

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

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

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

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

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

%F 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).

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

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

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

%Y Cf. A001620, A013661, A073002.

%K nonn,easy,mult

%O 1,2

%A _Amiram Eldar_, Aug 26 2023