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A162641 Number of even exponents in canonical prime factorization of n.

%I A162641 #25 Dec 25 2021 09:50:38
%S A162641 0,0,0,1,0,0,0,0,1,0,0,1,0,0,0,1,0,1,0,1,0,0,0,0,1,0,0,1,0,0,0,0,0,0,
%T A162641 0,2,0,0,0,0,0,0,0,1,1,0,0,1,1,1,0,1,0,0,0,0,0,0,0,1,0,0,1,1,0,0,0,1,
%U A162641 0,0,0,1,0,0,1,1,0,0,0,1,1,0,0,1,0,0,0,0,0,1,0,1,0,0,0,0,0,1,1,2,0,0,0,0,0
%N A162641 Number of even exponents in canonical prime factorization of n.
%H A162641 Antti Karttunen, <a href="/A162641/b162641.txt">Table of n, a(n) for n = 1..10000</a>
%H A162641 <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>.
%F A162641 a(n) = A001221(n) - A162642(n).
%F A162641 a(A002035(n)) = 0.
%F A162641 a(A072587(n)) > 0.
%F A162641 Additive with a(p^e) = A059841(e). - _Antti Karttunen_, Jul 23 2017
%F A162641 From _Antti Karttunen_, Nov 28 2017: (Start)
%F A162641 a(n) = A162642(A003557(n)).
%F A162641 a(n) <= A056170(n).
%F A162641 (End)
%F A162641 Asymptotic mean: lim_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{p prime} 1/(p*(p+1)) = 0.3302299262... (A179119). - _Amiram Eldar_, Dec 25 2021
%t A162641 Table[Count[FactorInteger[n][[All, -1]], _?EvenQ], {n, 105}] (* _Michael De Vlieger_, Jul 23 2017 *)
%o A162641 (PARI) A162641(n) = omega(n) - omega(core(n)); \\ _Antti Karttunen_, Jul 23 2017
%o A162641 (Scheme) (define (A162641 n) (if (= 1 n) 0 (+ (A059841 (A067029 n)) (A162641 (A028234 n))))) ;; _Antti Karttunen_, Jul 23 2017
%Y A162641 Cf. A001221, A003557, A002035, A007913, A025487, A059841, A061742 (where records occur), A072587, A162642, A179119.
%Y A162641 Cf. A268335 (positions of zeros), A295316.
%K A162641 nonn
%O A162641 1,36
%A A162641 _Reinhard Zumkeller_, Jul 08 2009