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).
FORMULA
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) <= 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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Reinhard Zumkeller, Jul 08 2009
STATUS
approved