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A350390 a(n) is the largest exponentially odd divisor of n.

%I A350390 #18 Sep 18 2023 02:02:10
%S A350390 1,2,3,2,5,6,7,8,3,10,11,6,13,14,15,8,17,6,19,10,21,22,23,24,5,26,27,
%T A350390 14,29,30,31,32,33,34,35,6,37,38,39,40,41,42,43,22,15,46,47,24,7,10,
%U A350390 51,26,53,54,55,56,57,58,59,30,61,62,21,32,65,66,67,34,69
%N A350390 a(n) is the largest exponentially odd divisor of n.
%C A350390 First differs from A331737 at n = 16.
%H A350390 Amiram Eldar, <a href="/A350390/b350390.txt">Table of n, a(n) for n = 1..10000</a>
%F A350390 Multiplicative with a(p^e) = p^e if e is odd and p^(e-1) otherwise.
%F A350390 a(n) = n/A336643(n).
%F A350390 a(n) = n if and only if n is an exponentially odd number (A268335).
%F A350390 Sum_{k=1..n} a(k) ~ (1/2)*c*n^2, where c = Product_{p prime} 1-(p-1)/(p^2*(p+1)) = 0.8073308216... (A306071).
%F A350390 Dirichlet g.f.: zeta(2*s-2) * Product_{p prime} (1 + 1/p^(s-1) - 1/p^(2*s-2) + 1/p^(2*s-1)). - _Amiram Eldar_, Sep 18 2023
%t A350390 f[p_, e_] := If[OddQ[e], p^e, p^(e - 1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A350390 (Python)
%o A350390 from math import prod
%o A350390 from sympy.ntheory.factor_ import primefactors, core
%o A350390 def A350390(n): return n*core(n)//prod(primefactors(n)) # _Chai Wah Wu_, Dec 30 2021
%o A350390 (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^(f[i,2] - !(f[i,2]%2)));} \\ _Amiram Eldar_, Sep 18 2023
%Y A350390 Cf. A268335, A306071, A325837, A331737, A336643, A350389.
%K A350390 nonn,easy,mult
%O A350390 1,2
%A A350390 _Amiram Eldar_, Dec 28 2021