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

1,2

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)

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

Mathematica

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

Prog

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

Crossrefs

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

Sequence in context: A295878 A368977 A294931 * A336856 A193292 A275297

Adjacent sequences: A365399 A365400 A365401 * A365403 A365404 A365405

Keyword

nonn,easy,mult

Author

Amiram Eldar, Sep 03 2023

Status

approved