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A325837 The number of coreful divisors of n that are exponentially odd numbers (A268335).

%I A325837 #47 Feb 11 2024 02:41:29
%S A325837 1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,2,1,1,2,1,1,1,1,3,1,1,
%T A325837 1,1,1,1,1,2,1,1,1,1,1,1,1,2,1,1,1,1,1,2,1,2,1,1,1,1,1,1,1,3,1,1,1,1,
%U A325837 1,1,1,2,1,1,1,1,1,1,1,2,2,1,1,1,1,1,1
%N A325837 The number of coreful divisors of n that are exponentially odd numbers (A268335).
%C A325837 First differs from A050361 at n = 64.
%C A325837 From _Amiram Eldar_, Sep 08 2023: (Start)
%C A325837 The number of exponentially odd divisors of n is A322483(n), and their sum is A033634(n).
%C A325837 A coreful divisor d of a number n is a divisor with the same set of distinct prime factors as n. (End)
%C A325837 Also, the number of divisors of n that are cubefull exponentially odd numbers (A335988). - _Amiram Eldar_, Feb 11 2024
%H A325837 Amiram Eldar, <a href="/A325837/b325837.txt">Table of n, a(n) for n = 1..10000</a>
%F A325837 Multiplicative with a(p^e) = floor((e+1)/2).
%F A325837 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + 1/(p*(p^2-1))) = 1.231291... (A065487). - _Amiram Eldar_, Sep 10 2022
%F A325837 a(n) = A046951(A350390(n)) (the number of squares dividing the largest exponentially odd divisor of n). - _Amiram Eldar_, Sep 01 2023
%F A325837 From _Amiram Eldar_, Sep 08 2023: (Start)
%F A325837 a(n) = A046951(A003557(n)).
%F A325837 Dirichlet g.f.: zeta(s) * zeta(2*s) * Product_{p prime} (1 - 1/p^(2*s) + 1/p^(3*s)). (End)
%t A325837 fun[p_,e_] := Floor[(e+1)/2]; a[n_] := Times@@(fun@@@FactorInteger[n]); Array[a, 100]
%o A325837 (PARI) a(n) = vecprod(apply(x -> (x+1)\2, factor(n)[, 2])); \\ _Amiram Eldar_, Sep 01 2023
%Y A325837 Cf. A033634, A050361, A065487, A268335, A295316, A335988, A350390.
%Y A325837 Cf. A003557, A005361 (number of coreful divisors), A046951, A268335.
%K A325837 nonn,easy,mult
%O A325837 1,8
%A A325837 _Amiram Eldar_, Sep 07 2019
%E A325837 Name corrected by _Amiram Eldar_, Sep 08 2023