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A365402
The number of divisors of the largest unitary divisor of n that is an exponentially odd number.
3
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).
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
KEYWORD
nonn,easy,mult
AUTHOR
Amiram Eldar, Sep 03 2023
STATUS
approved