cp's OEIS Frontend

a(n) = A000005(A350388(n)).

a(n) = A000005(n) / A365402(n).

a(n) <= A000005(n) with equality if and only if n is a square (A000290).

a(n) >= 1 with equality if and only if n is an exponentially odd number (A268335).

Multiplicative with a(p^e) = 1 if e is odd, and e+1 if e is even.

Dirichlet g.f.: zeta(2*s)^2 * Product_{p prime} (1 + 1/p^s + 1/p^(2*s) - 1/p^(3*s)).

From Vaclav Kotesovec, Sep 05 2023: (Start)

Dirichlet g.f.: zeta(s) * zeta(2*s)^2 * Product_{p prime} (1 - 2/p^(3*s) + 1/p^(4*s)).

Let f(s) = Product_{p prime} (1 - 2/p^(3*s) + 1/p^(4*s)).

Sum_{k=1..n} a(k) ~ f(1) * Pi^4 * n / 36 + sqrt(n) * zeta(1/2) * f(1/2)/2 * (log(n) + 4*gamma - 2 + zeta'(1/2)/zeta(1/2) + f'(1/2)/f(1/2)), where

f(1) = Product_{p prime} (1 - 2/p^3 + 1/p^4) = 0.7446954979060674204391238715944543281179691329049241118630718137015097502...,

f(1/2) = Product_{p prime} (1 - 2/p^(3/2) + 1/p^2) = 0.2312522106782016049013780988087017618011735848676872392115785564006277675...,

f'(1/2) = f(1/2) * Sum_{p prime} 2*(3*sqrt(p) - 2) * log(p) / (1 - 2*sqrt(p) + p^2) = f(1/2) * 6.937179176924511608542644054340717439502789953858512457656... and gamma is the Euler-Mascheroni constant A001620. (End)

Multiplicative with a(p^e) = A048298(e) + 1.

a(n) = A000005(A367168(n)).

a(n) <= A000005(n), with equality if and only if n is in A138302.

a(n) = A000010(A350389(n)).

a(n) >= 1, with equality if and only if n is either a square (A000290) or twice and odd square (A077591 \ {1}).

a(n) <= A000010(n), with equality if and only if n is an exponentially odd number (A268335).

Multiplicative with a(p^e) = (p-1)*p^(e-1) if e is odd, and 1 otherwise.

Dirichlet g.f.: zeta(2*s-2) * zeta(2*s) * Product_{p prime} (1 - 1/p^s + 1/p^(s-1) - 1/p^(2*s-2) - 1/p^(3*s-1) + 1/p^(3*s)).

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = zeta(4) * Product_{p prime} (1 - 2/p^2 + 2/p^3 - 2/p^4 + 1/p^5) = 0.50115112192510092436... .