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A046645 a(n) = log_2(A046644(n)); also the 2-adic valuation of A046644(n).

%I A046645 #30 Sep 29 2023 04:03:57
%S A046645 0,1,1,3,1,2,1,4,3,2,1,4,1,2,2,7,1,4,1,4,2,2,1,5,3,2,4,4,1,3,1,8,2,2,
%T A046645 2,6,1,2,2,5,1,3,1,4,4,2,1,8,3,4,2,4,1,5,2,5,2,2,1,5,1,2,4,10,2,3,1,4,
%U A046645 2,3,1,7,1,2,4,4,2,3,1,8,7,2,1,5,2,2,2,5,1,5,2,4,2
%N A046645 a(n) = log_2(A046644(n)); also the 2-adic valuation of A046644(n).
%C A046645 A268375 gives numbers n for which a(n) = A289617(n) = A005187(A001222(n)). - _Antti Karttunen_, Jul 08 2017
%H A046645 Antti Karttunen, <a href="/A046645/b046645.txt">Table of n, a(n) for n = 1..65536</a>
%H A046645 <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>.
%F A046645 a(n) = A007814(A046644(n)). - _Michel Marcus_, Apr 16 2015
%F A046645 Additive with a(p^n) = A005187(n). - _Antti Karttunen_, Jul 08 2017
%F A046645 a(n) = A293447(A293442(n)). - _Antti Karttunen_, Nov 10 2017
%F A046645 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) = 1.410258867603361890498..., where f(x) = -x + Sum_{k>=0} (2^(k+1)-1)*x^(2^k)/(1+x^(2^k)). - _Amiram Eldar_, Sep 29 2023
%t A046645 f[p_, e_] := 2*e - DigitCount[2*e, 2, 1]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Apr 28 2023 *)
%o A046645 (PARI)
%o A046645 A007814(n) = (valuation(n,2));
%o A046645 A046643perA046644(n) = { my(c=1); if(1==n,c,fordiv(n,d, if((d>1)&&(d<n), c -= (A046643perA046644(d)*A046643perA046644(n/d)))); (c/2)); } \\ After the Maple-program given in A046643.
%o A046645 A046645(n) = A007814(denominator(A046643perA046644(n))); \\ _Antti Karttunen_, Jul 08 2017
%o A046645 (PARI)
%o A046645 A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
%o A046645 A046645(n) = vecsum(apply(e -> A005187(e), factorint(n)[, 2])); \\ A faster implementation. - _Antti Karttunen_, Jul 08 2017
%Y A046645 See A046643, A046644 for more details.
%Y A046645 Cf. A001222, A005187, A007814, A077761, A268375, A289617, A289618, A293442, A293447.
%K A046645 nonn,easy
%O A046645 1,4
%A A046645 _N. J. A. Sloane_