%I #16 Sep 30 2023 21:56:48
%S 0,0,1,1,1,1,1,1,3,1,1,2,1,1,2,3,1,3,1,2,2,1,1,2,3,1,4,2,1,2,1,3,2,1,
%T 2,4,1,1,2,2,1,2,1,2,4,1,1,4,3,3,2,2,1,4,2,2,2,1,1,3,1,1,4,4,2,2,1,2,
%U 2,2,1,4,1,1,4,2,2,2,1,4,7,1,1,3,2,1,2,2,1,4,2,2,2,1,2,4,1,3,4,4,1,2,1,2,3
%N a(n) = A046645(n) - A007814(n); the 2-adic valuation of A299150.
%C After two initial terms, all terms are positive.
%H Antti Karttunen, <a href="/A318440/b318440.txt">Table of n, a(n) for n = 1..65537</a>
%F a(n) = A046645(n) - A007814(n).
%F a(n) = A007814(A299150(n)).
%F Additive with a(p^e) = (1 + (p mod 2))*e - A000120(e). - _Amiram Eldar_, Apr 28 2023
%F Sum_{k=1..n} a(k) ~ n * (log(log(n)) + B + C), where B is Mertens's constant (A077761) and C = -1 + Sum_{p prime} f(1/p) = 0.410258867603361890498..., where f(x) = -x + Sum_{k>=0} (2^(k+1)-1)*x^(2^k)/(1+x^(2^k)). - _Amiram Eldar_, Sep 30 2023
%t f[p_, e_] := (1 + Mod[p, 2])*e - DigitCount[e, 2, 1]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Apr 28 2023 *)
%o (PARI)
%o A007814(n) = valuation(n,2);
%o A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
%o A046645(n) = vecsum(apply(e -> A005187(e),factor(n)[,2]));
%o A318440(n) = A046645(n) - A007814(n);
%Y Cf. A000120, A007814, A046645, A077761, A299150, A318656.
%Y Cf. also A305439.
%K nonn,easy
%O 1,9
%A _Antti Karttunen_, Sep 02 2018