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A372503 The number of prime powers that are noninfinitary divisors of n.

%I A372503 #7 May 04 2024 05:35:36
%S A372503 0,0,0,1,0,0,0,0,1,0,0,1,0,0,0,3,0,1,0,1,0,0,0,0,1,0,0,1,0,0,0,2,0,0,
%T A372503 0,2,0,0,0,0,0,0,0,1,1,0,0,3,1,1,0,1,0,0,0,0,0,0,0,1,0,0,1,3,0,0,0,1,
%U A372503 0,0,0,1,0,0,1,1,0,0,0,3,3,0,0,1,0,0,0
%N A372503 The number of prime powers that are noninfinitary divisors of n.
%C A372503 First differs from A318499 at n = 32.
%H A372503 Amiram Eldar, <a href="/A372503/b372503.txt">Table of n, a(n) for n = 1..10000</a>
%F A372503 Additive with a(p^e) = e - 2^A000120(e) + 1 = A048967(e).
%F A372503 a(n) = A001222(n) - A349258(n).
%F A372503 a(n) = 0 if and only if n is in A036537.
%F A372503 a(n) > 0 if and only if n is in A162643.
%F A372503 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{p prime} f(1/p) = 0.4917971717413486467..., where f(x) = 1/(1-x) - (1-x) * Product_{k>=0} (1 + 2*x^(2^k)).
%t A372503 f[p_, e_] :=  e + 1 - 2^DigitCount[e, 2, 1]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A372503 (PARI) a(n) = vecsum(apply(x -> x + 1 - 1 << hammingweight(x), factor(n)[, 2]));
%Y A372503 Cf. A000120, A001222, A036537, A048967, A162643, A318499, A349258.
%K A372503 nonn,easy
%O A372503 1,16
%A A372503 _Amiram Eldar_, May 04 2024