A365402 The number of divisors of the largest unitary divisor of n that is an exponentially odd number.

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

Offset: 1

Author

Amiram Eldar, Sep 03 2023

Keywords

• nonn
• easy
• mult

Comments

The sum of these divisors is A351569(n).

All the terms are either 1 or even (A004277).

Links

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

Crossrefs

Cf. A000005, A000290, A004277, A268335, A350389, A351569, A365401.

Programs

Mathematica

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

PARI

a(n) = vecprod(apply(x -> if(x%2, x+1, 1), factor(n)[, 2]));

Python

from math import prod
from sympy import factorint
def A365402(n): return prod(e+1 for e in factorint(n).values() if e&1) # Chai Wah Wu, Nov 17 2023

SageMath

def a(n): return prod((valuation(n,p)+1) for p in prime_divisors(n) if valuation(n,p)%2==1) # Orges Leka, Nov 16 2023

Formula

a(n) = A000005(A350389(n)).

a(n) = A000005(n) / A365401(n).

a(n) <= A000005(n) with equality if and only if n is an exponentially odd number (A268335).

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

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

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

From Vaclav Kotesovec, Sep 05 2023: (Start)

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

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

Sum_{k=1..n} a(k) ~ f(1) * Pi^4 * n / 36 * (log(n) + 2*gamma - 1 + 24*Zeta'(2)/Pi^2 + f'(1)/f(1)), where

f(1) = Product_{p prime} (1 - 4/p^2 + 4/p^3 - 1/p^4) = 0.2177787166195363783230075141194468131307977550013559376482764035236264911...

f'(1) = f(1) * Sum_{p prime} 4*(2*p - 1) * log(p) / (1 - 3*p + p^2 + p^3) = f(1) * 3.3720882314412399056794495057358594564001229865925330149186567502684770675...

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

a(n) = Sum_{d|n} (-1)^(Sum_{p|gcd(d,n/d)} v_p(d)*v_p(n/d)), where v_p(x) denotes the valuation of x at the prime p. - Orges Leka, Nov 16 2023