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A350387 a(n) is the sum of the odd exponents in the prime factorization of n.

%I A350387 #17 Oct 07 2023 09:51:07
%S A350387 0,1,1,0,1,2,1,3,0,2,1,1,1,2,2,0,1,1,1,1,2,2,1,4,0,2,3,1,1,3,1,5,2,2,
%T A350387 2,0,1,2,2,4,1,3,1,1,1,2,1,1,0,1,2,1,1,4,2,4,2,2,1,2,1,2,1,0,2,3,1,1,
%U A350387 2,3,1,3,1,2,1,1,2,3,1,1,0,2,1,2,2,2,2,4,1,2,2,1,2,2,2,6,1,1,1,0,1,3,1,4,3
%N A350387 a(n) is the sum of the odd exponents in the prime factorization of n.
%C A350387 First differs from A125073 at n = 32.
%C A350387 a(n) is the number of prime divisors of n, counted with multiplicity, with an odd exponent in the prime factorization of n.
%H A350387 Amiram Eldar, <a href="/A350387/b350387.txt">Table of n, a(n) for n = 1..10000</a>
%F A350387 Additive with a(p^e) = e if e is odd and 0 otherwise.
%F A350387 a(n) = A001222(A350389(n)).
%F A350387 a(n) = 0 if and only if n is a positive square (A000290 \ {0}).
%F A350387 a(n) = A001222(n) - A350386(n).
%F A350387 a(n) = A001222(n) if and only if n is an exponentially odd number (A268335).
%F A350387 Sum_{k=1..n} a(k) = n * log(log(n)) + c * n + O(n/log(n)), where c = A083342 - Sum_{p prime} 2*p/((p-1)*(p+1)^2) = gamma + Sum_{p prime} (log(1-1/p) + (p^2+1)/((p-1)*(p+1)^2)) = 0.2384832800... and gamma is Euler's constant (A001620).
%t A350387 f[p_, e_] := If[OddQ[e], e, 0]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A350387 (Python)
%o A350387 from sympy import factorint
%o A350387 def a(n): return sum(e for e in factorint(n).values() if e%2 == 1)
%o A350387 print([a(n) for n in range(1, 106)]) # _Michael S. Branicky_, Dec 28 2021
%o A350387 (PARI) a(n) = my(f=factor(n)); sum(k=1, #f~, if (f[k,2] %2, f[k,2])); \\ _Michel Marcus_, Dec 28 2021
%Y A350387 Cf. A000290, A001222, A001620, A083342, A125073, A162641, A268335, A350386, A350389.
%K A350387 nonn,easy
%O A350387 1,6
%A A350387 _Amiram Eldar_, Dec 28 2021