A368470 a(n) is the number of exponentially odd divisors of the largest unitary divisor of n that is an exponentially odd number (A268335).

1, 2, 2, 1, 2, 4, 2, 3, 1, 4, 2, 2, 2, 4, 4, 1, 2, 2, 2, 2, 4, 4, 2, 6, 1, 4, 3, 2, 2, 8, 2, 4, 4, 4, 4, 1, 2, 4, 4, 6, 2, 8, 2, 2, 2, 4, 2, 2, 1, 2, 4, 2, 2, 6, 4, 6, 4, 4, 2, 4, 2, 4, 2, 1, 4, 8, 2, 2, 4, 8, 2, 3, 2, 4, 2, 2, 4, 8, 2, 2, 1, 4, 2, 4, 4, 4, 4

Offset: 1

Author

Amiram Eldar, Dec 26 2023

Keywords

• nonn
• easy
• mult

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)

Crossrefs

Cf. A000005, A000290, A005117, A033634, A268335, A350389, A368471.

Programs

Mathematica

f[p_, e_] := If[OddQ[e], (e + 3)/2, 1]; 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,2]%2, (f[i,2]+3)/2, 1));}

Formula

a(n) = A033634(A350389(n)).

Multiplicative with a(p^e) = (e+3)/2 if e is odd and 1 otherwise.

a(n) >= 1, with equality if and only if n is a square (A000290).

a(n) <= A000005(n), with equality if and only if n is squarefree (A005117).

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

From Vaclav Kotesovec, Dec 26 2023: (Start)

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

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

Sum_{k=1..n} a(k) ~ f(1) * n * (log(n) + 2*gamma - 1 + f'(1)/f(1)), where

f(1) = Product_{p prime} (1 - (2*p+1) / (p*(p+1)^2)) = 0.528940778823659679133966695786017426052491935740673837882972347697...,

f'(1) = f(1) * Sum_{p prime} (4*p^2 + 3*p + 1) * log(p) / (p^4 + 3*p^3 + p^2 - 2*p - 1) = f(1) * 1.36109933267802415215189866467122940932493907539386280428818...

and gamma is the Euler-Mascheroni constant A001620. (End)