A369307 - OEIS

%I #15 Jan 20 2024 03:09:34

%S 1,2,2,1,2,4,2,2,1,4,2,2,2,4,4,2,2,2,2,2,4,4,2,4,1,4,2,2,2,8,2,2,4,4,

%T 4,1,2,4,4,4,2,8,2,2,2,4,2,4,1,2,4,2,2,4,4,4,4,4,2,4,2,4,2,3,4,8,2,2,

%U 4,8,2,2,2,4,2,2,4,8,2,4,2,4,2,4,4,4,4

%N The number of exponentially odd divisors d of n such that n/d is also exponentially odd.

%C First differs from A366308 at n = 32.

%C Dirichlet convolution of A295316 with itself.

%H Amiram Eldar, <a href="/A369307/b369307.txt">Table of n, a(n) for n = 1..10000</a>

%H Vaclav Kotesovec, <a href="/A369307/a369307.jpg">Graph - the asymptotic ratio (100000 terms)</a>

%F Multiplicative with a(p^e) = 2 is e is odd, and e/2 if e is even.

%F a(n) >= 1, with equality if and only if n is the square of a squarefree number (A062503).

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

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

%F From _Vaclav Kotesovec_, Jan 19 2024: (Start)

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

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

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

%F f(1) = Product_{p prime} (1 - (2*p^2 + 2*p - 1) / (p^2*(p+1)^2)) = 0.49623881454854881762168565097162197963340069996226074849602334089041678...,

%F f'(1) = f(1) * Sum_{p prime} 2*(2*p + 1) * log(p) / ((p+1)*(p^2 + p - 1)) = f(1) * 1.49674466685934940187617305887881799198585080518913793200171026177150513...

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

%t f[p_,e_] := If[OddQ[e], 2, e/2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

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

%o (PARI) for(n=1, 100, print1(direuler(p=2, n, (1 - X^2 + X)^2/(1 - X^2)^2)[n], ", ")) \\ _Vaclav Kotesovec_, Jan 19 2024

%o (Python)

%o from math import prod

%o from sympy import factorint

%o def A369307(n): return prod(2 if e&1 else e>>1 for e in factorint(n).values()) # _Chai Wah Wu_, Jan 19 2024

%Y Cf. A000005, A005117, A062503, A268335, A295316, A322483, A365173, A366308, A369308.

%K nonn,easy,mult

%O 1,2

%A _Amiram Eldar_, Jan 19 2024