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

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

dpf[n_] := Module[{df = PrimeNu[n]}, If[OddQ[df], 1, -1]]; Join[{0}, Accumulate[ Array[dpf, 100, 2]]] (* Harvey P. Dale, Jul 04 2012 *)

Prog

(Python)
from sympy import primenu
def A123066(n): return 1+sum(1 if primenu(i)&1 else -1 for i in range(1, n+1)) # Chai Wah Wu, Dec 31 2022

Crossrefs

Cf. A030230, A030231.

Cf. A072203, A001221.

Cf. A346617.

Sequence in context: A262519 A225320 A308937 * A330239 A235121 A270652

Adjacent sequences: A123063 A123064 A123065 * A123067 A123068 A123069

Keyword

easy,look,sign

Author

Franklin T. Adams-Watters, Sep 26 2006

Status

approved