cp's OEIS Frontend

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

%I A365402 #19 Nov 17 2023 21:34:33
%S A365402 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,
%T A365402 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,
%U A365402 4,8,2,4,2,4,2,2,4,8,2,2,1,4,2,4,4,4,4
%N A365402 The number of divisors of the largest unitary divisor of n that is an exponentially odd number.
%C A365402 The sum of these divisors is A351569(n).
%C A365402 All the terms are either 1 or even (A004277).
%H A365402 Amiram Eldar, <a href="/A365402/b365402.txt">Table of n, a(n) for n = 1..10000</a>
%H A365402 Vaclav Kotesovec, <a href="/A365402/a365402.jpg">Graph - the asymptotic ratio (100000 terms)</a>
%F A365402 a(n) = A000005(A350389(n)).
%F A365402 a(n) = A000005(n) / A365401(n).
%F A365402 a(n) <= A000005(n) with equality if and only if n is an exponentially odd number (A268335).
%F A365402 a(n) >= 1 with equality if and only if n is a square (A000290).
%F A365402 Multiplicative with a(p^e) = 1 if e is even, and e+1 if e is odd.
%F A365402 Dirichlet g.f.: zeta(2*s)^2 * Product_{p prime} (1 + 2/p^s - 1/p^(2*s)).
%F A365402 From _Vaclav Kotesovec_, Sep 05 2023: (Start)
%F A365402 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)).
%F A365402 Let f(s) = Product_{p prime} (1 - 4/p^(2*s) + 4/p^(3*s) - 1/p^(4*s)).
%F A365402 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 A365402 f(1) = Product_{p prime} (1 - 4/p^2 + 4/p^3 - 1/p^4) = 0.2177787166195363783230075141194468131307977550013559376482764035236264911...
%F A365402 f'(1) = f(1) * Sum_{p prime} 4*(2*p - 1) * log(p) / (1 - 3*p + p^2 + p^3) = f(1) * 3.3720882314412399056794495057358594564001229865925330149186567502684770675...
%F A365402 and gamma is the Euler-Mascheroni constant A001620. (End)
%F A365402 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
%t A365402 f[p_, e_] := If[OddQ[e], e + 1, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A365402 (PARI) a(n) = vecprod(apply(x -> if(x%2, x+1, 1), factor(n)[, 2]));
%o A365402 (SageMath)
%o A365402 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
%o A365402 (Python)
%o A365402 from math import prod
%o A365402 from sympy import factorint
%o A365402 def A365402(n): return prod(e+1 for e in factorint(n).values() if e&1) # _Chai Wah Wu_, Nov 17 2023
%Y A365402 Cf. A000005, A000290, A004277, A268335, A350389, A351569, A365401.
%K A365402 nonn,easy,mult
%O A365402 1,2
%A A365402 _Amiram Eldar_, Sep 03 2023