A162642 Number of odd exponents in the canonical prime factorization of n.

0, 1, 1, 0, 1, 2, 1, 1, 0, 2, 1, 1, 1, 2, 2, 0, 1, 1, 1, 1, 2, 2, 1, 2, 0, 2, 1, 1, 1, 3, 1, 1, 2, 2, 2, 0, 1, 2, 2, 2, 1, 3, 1, 1, 1, 2, 1, 1, 0, 1, 2, 1, 1, 2, 2, 2, 2, 2, 1, 2, 1, 2, 1, 0, 2, 3, 1, 1, 2, 3, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 0, 2, 1, 2, 2, 2, 2, 2, 1, 2, 2, 1, 2, 2, 2, 2, 1, 1, 1, 0, 1, 3, 1, 2, 3

Offset

1,6

Comments

a(n) is also known as the squarefree rank of n. - Jason Kimberley, Jul 08 2017

The number of primes that are infinitary divisors of n. - Amiram Eldar, Oct 01 2023

Links

Jason Kimberley, Table of n, a(n) for n = 1..20000

R. B. Eggleton, J. S. Kimberley and J. A. MacDougall, Square-free rank of integers, Journal of Combinatorial Mathematics and Combinatorial Computing, Vol. 106 (2018).

Index entries for sequences computed from exponents in factorization of n.

Formula

a(n) = A001221(n) - A162641(n).

a(n) = A001221(A007913(n)). - Jason Kimberley, Jan 06 2016

a(A000290(n)) = 0, n > 0. - Michel Marcus, Jan 08 2016

G.f.: Sum_{i>=1} Sum_{j>=1} (-1)^j x^(prime(i)^j)/(x^(prime(i)^j) - 1). - Robert Israel, Jan 15 2016

From Antti Karttunen, Nov 28 2017: (Start)

Additive with a(p^e) = A000035(e).

a(n) = A056169(n) + A295662(n).

A056169(n) <= a(n) <= A056169(n) + A295659(n).

a(n) <= A295664(n).

(End)

Sum_{k=1..n} a(k) = n * log(log(n)) + c * n + O(n/log(n)), where c = gamma + Sum_{p prime} (log(1-1/p) + 1/(p+1)) = A077761 - A179119 = -0.0687327134... and gamma is Euler's constant (A001620). - Amiram Eldar, Dec 25 2021

Maple

A162642 := proc(n) add ( op(2, f) mod 2 , f=ifactors(n)[2]) ; end proc: # R. J. Mathar, Mar 30 2011

Mathematica

{0}~Join~Table[Count[Last /@ FactorInteger@ n, e_ /; OddQ@ e], {n, 2, 105}] (* Michael De Vlieger, Jan 06 2016 *)

Prog

(Magma) A162642:=func<n|#{pe:pe in Factorisation(n)|IsOdd(pe[2])}>;
[A162642(n):n in[1..105]]; // Jason Kimberley, Dec 30 2015
(PARI) a(n) = {my(f = factor(n)); sum(k=1, #f~, f[k, 2] % 2); } \\ Michel Marcus, Jan 08 2016
(Scheme, with memoization-macro definec) (definec (A162642 n) (if (= 1 n) 0 (+ (A000035 (A067029 n)) (A162642 (A028234 n))))) ;; Antti Karttunen, Nov 28 2017

Crossrefs

Cf. A000290 (positions of zeros), A001221, A001620, A002035, A007913, A056169, A162641, A295316, A295659, A295662, A295664.

Sequence in context: A255315 A373029 A125072 * A366246 A361205 A355827

Adjacent sequences: A162639 A162640 A162641 * A162643 A162644 A162645

Keyword

nonn,easy

Author

Reinhard Zumkeller, Jul 08 2009

Status

approved