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The number of prime powers that are noninfinitary divisors of n.
1

%I #7 May 04 2024 05:35:36

%S 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 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 0,0,0,1,0,0,1,1,0,0,0,3,3,0,0,1,0,0,0

%N The number of prime powers that are noninfinitary divisors of n.

%C First differs from A318499 at n = 32.

%H Amiram Eldar, <a href="/A372503/b372503.txt">Table of n, a(n) for n = 1..10000</a>

%F Additive with a(p^e) = e - 2^A000120(e) + 1 = A048967(e).

%F a(n) = A001222(n) - A349258(n).

%F a(n) = 0 if and only if n is in A036537.

%F a(n) > 0 if and only if n is in A162643.

%F 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 f[p_, e_] := e + 1 - 2^DigitCount[e, 2, 1]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100]

%o (PARI) a(n) = vecsum(apply(x -> x + 1 - 1 << hammingweight(x), factor(n)[, 2]));

%Y Cf. A000120, A001222, A036537, A048967, A162643, A318499, A349258.

%K nonn,easy

%O 1,16

%A _Amiram Eldar_, May 04 2024