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A295664 Exponent of the highest power of 2 dividing number of divisors of n: a(n) = A007814(A000005(n)); 2-adic valuation of tau(n).

%I A295664 #31 Sep 28 2023 04:21:29
%S A295664 0,1,1,0,1,2,1,2,0,2,1,1,1,2,2,0,1,1,1,1,2,2,1,3,0,2,2,1,1,3,1,1,2,2,
%T A295664 2,0,1,2,2,3,1,3,1,1,1,2,1,1,0,1,2,1,1,3,2,3,2,2,1,2,1,2,1,0,2,3,1,1,
%U A295664 2,3,1,2,1,2,1,1,2,3,1,1,0,2,1,2,2,2,2,3,1,2,2,1,2,2,2,2,1,1,1,0,1,3,1,3,3,2,1,2,1,3,2,1,1,3,2,1,1,2,2,4,0
%N A295664 Exponent of the highest power of 2 dividing number of divisors of n: a(n) = A007814(A000005(n)); 2-adic valuation of tau(n).
%C A295664 In the prime factorization of n = p1^e1 * ... pk^ek, add together the number of trailing 1-bits in each exponent e when they are written in binary.
%H A295664 Antti Karttunen, <a href="/A295664/b295664.txt">Table of n, a(n) for n = 1..65537</a>
%H A295664 <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>
%F A295664 Additive with a(p^e) = A007814(1+e).
%F A295664 a(1) = 0; for n > 1, a(n) = A007814(1+A067029(n)) + a(A028234(n)).
%F A295664 a(n) = A007814(A000005(n)).
%F A295664 a(n) >= A162642(n) >= A056169(n).
%F A295664 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) =-0.223720656976344505701..., where f(x) = -x + (1-x) * Sum_{k>=1} x^(2^k-1)/(1-x^(2^k)). - _Amiram Eldar_, Sep 28 2023
%t A295664 Table[IntegerExponent[DivisorSigma[0, n], 2], {n, 120}] (* _Michael De Vlieger_, Nov 28 2017 *)
%o A295664 (Scheme, with memoization-macro definec)
%o A295664 (definec (A295664 n) (if (= 1 n) 0 (+ (A007814 (+ 1 (A067029 n))) (A295664 (A028234 n)))))
%o A295664 (define (A295664 n) (A007814 (A000005 n)))
%o A295664 (PARI) a(n) = valuation(numdiv(n), 2); \\ _Michel Marcus_, Nov 30 2017
%o A295664 (Python)
%o A295664 from sympy import divisor_count
%o A295664 def A295664(n): return (~(m:=int(divisor_count(n))) & m-1).bit_length() # _Chai Wah Wu_, Jul 05 2022
%Y A295664 Cf. A000005, A007814, A028234, A056169, A067029, A077761, A162642, A295663.
%Y A295664 Cf. A000290 (positions of zeros).
%K A295664 nonn,easy
%O A295664 1,6
%A A295664 _Antti Karttunen_, Nov 28 2017