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

Author

Reinhard Zumkeller, Jul 08 2009

Keywords

• nonn
• easy

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.

Crossrefs

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

Programs

Magma

A162642:=func;
[A162642(n):n in[1..105]]; // Jason Kimberley, Dec 30 2015

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 *)

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

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