A123066 - OEIS

A123066

(Number of numbers <= n with an odd number of distinct prime factors) - (number of numbers <= n with an even number of distinct prime factors).

0, 1, 2, 3, 4, 3, 4, 5, 6, 5, 6, 5, 6, 5, 4, 5, 6, 5, 6, 5, 4, 3, 4, 3, 4, 3, 4, 3, 4, 5, 6, 7, 6, 5, 4, 3, 4, 3, 2, 1, 2, 3, 4, 3, 2, 1, 2, 1, 2, 1, 0, -1, 0, -1, -2, -3, -4, -5, -4, -3, -2, -3, -4, -3, -4, -3, -2, -3, -4, -3, -2, -3, -2, -3, -4, -5, -6, -5, -4, -5, -4, -5, -4, -3, -4, -5, -6, -7, -6, -5, -6, -7, -8, -9, -10

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

Analog of A072203 for number of distinct factors. Conjecture that sequence changes sign infinitely often, although the next sign change is probably large.

The signs first change at n = 52 and then change again at n = 7954. - Harvey P. Dale, Jul 04 2012

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..5000

H. Helfgott and A. Ubis, Primos, paridad y análisis, arXiv:1812.08707 [math.NT], Dec. 2018.

FORMULA

a(n) = Sum_{k>=1} (-1)^(k-1) * A346617(n,k). - Alois P. Heinz, Aug 19 2021

MAPLE

a:= proc(n) option remember; `if`(n<2, 0, a(n-1)+

`if`(nops(ifactors(n)[2])::odd, 1, -1))

end:

seq(a(n), n=1..120); # Alois P. Heinz, Dec 21 2018

MATHEMATICA