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).

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, 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, 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

Offset

1,6

Comments

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.

Links

Antti Karttunen, Table of n, a(n) for n = 1..65537

Index entries for sequences computed from exponents in factorization of n

Formula

Additive with a(p^e) = A007814(1+e).

a(1) = 0; for n > 1, a(n) = A007814(1+A067029(n)) + a(A028234(n)).

a(n) = A007814(A000005(n)).

a(n) >= A162642(n) >= A056169(n).

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

Mathematica

Table[IntegerExponent[DivisorSigma[0, n], 2], {n, 120}] (* Michael De Vlieger, Nov 28 2017 *)

Prog

(Scheme) ;; With memoization-macro definec.
(definec (A295664 n) (if (= 1 n) 0 (+ (A007814 (+ 1 (A067029 n))) (A295664 (A028234 n)))))
(define (A295664 n) (A007814 (A000005 n)))
(PARI) a(n) = valuation(numdiv(n), 2); \\ Michel Marcus, Nov 30 2017
(Python)
from sympy import divisor_count
def A295664(n): return (~(m:=int(divisor_count(n))) & m-1).bit_length() # Chai Wah Wu, Jul 05 2022

Crossrefs

Cf. A000005, A007814, A028234, A056169, A067029, A077761, A162642, A295663.

Cf. A000290 (positions of zeros).

Sequence in context: A071460 A369862 A218867 * A250213 A033794 A218856

Adjacent sequences: A295661 A295662 A295663 * A295665 A295666 A295667

Keyword

nonn,easy

Author

Antti Karttunen, Nov 28 2017

Status

approved