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A349258 a(n) is the number of prime powers (not including 1) that are infinitary divisors of n.

%I A349258 #22 Sep 29 2023 04:05:56
%S A349258 0,1,1,1,1,2,1,3,1,2,1,2,1,2,2,1,1,2,1,2,2,2,1,4,1,2,3,2,1,3,1,3,2,2,
%T A349258 2,2,1,2,2,4,1,3,1,2,2,2,1,2,1,2,2,2,1,4,2,4,2,2,1,3,1,2,2,3,2,3,1,2,
%U A349258 2,3,1,4,1,2,2,2,2,3,1,2,1,2,1,3,2,2,2,4,1,3,2,2,2,2,2,4,1,2,2,2,1,3,1,4,3
%N A349258 a(n) is the number of prime powers (not including 1) that are infinitary divisors of n.
%C A349258 The total number of prime powers (not including 1) that divide n is A001222(n).
%C A349258 For each n, all the prime powers that are infinitary divisors of n are "Fermi-Dirac primes" (A050376).
%H A349258 Amiram Eldar, <a href="/A349258/b349258.txt">Table of n, a(n) for n = 1..10000</a>
%F A349258 Additive with a(p^e) = 2^A000120(e) - 1.
%F A349258 a(n) <= A001222(n), with equality if and only if n is in A036537.
%F A349258 a(n) <= A037445(n) - 1, with equality if and only if n is a prime power (including 1, A000961).
%F A349258 Sum_{k=1..n} a(k) ~ n * (log(log(n)) + B + C), where B is Mertens's constant (A077761) and C = Sum_{p prime} f(1/p) = 0.28135949730844648114..., where f(x) = -(x+1) + (1-x) * Product_{k>=0} (1 + 2*x^(2^k)). - _Amiram Eldar_, Sep 29 2023
%e A349258 12 has 4 infinitary divisors, 1, 3, 4 and 12. Two of these divisors, 3 and 4 = 2^2 are prime powers. Therefore a(12) = 2.
%t A349258 f[p_,e_] := 2^DigitCount[e, 2, 1] - 1; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a,100]
%o A349258 (PARI) A349258(n) = if(1==n,0,vecsum(apply(x->(2^hammingweight(x))-1,factor(n)[,2]))); \\ _Antti Karttunen_, Nov 12 2021
%Y A349258 Cf. A000120, A000961, A001222, A036537, A037445, A050376, A077609, A077761, A246655.
%K A349258 nonn,easy
%O A349258 1,6
%A A349258 _Amiram Eldar_, Nov 12 2021
%E A349258 Wrong comment removed by _Amiram Eldar_, Sep 22 2023