A248211 First differences of omega(n), the number of distinct prime factors function (A001221).

1, 0, 0, 0, 1, -1, 0, 0, 1, -1, 1, -1, 1, 0, -1, 0, 1, -1, 1, 0, 0, -1, 1, -1, 1, -1, 1, -1, 2, -2, 0, 1, 0, 0, 0, -1, 1, 0, 0, -1, 2, -2, 1, 0, 0, -1, 1, -1, 1, 0, 0, -1, 1, 0, 0, 0, 0, -1, 2, -2, 1, 0, -1, 1, 1, -2, 1, 0, 1, -2, 1, -1, 1, 0, 0, 0, 1, -2, 1

Offset: 1

Author

Wesley Ivan Hurt, Oct 04 2014

Keywords

• sign
• easy

Comments

First instance of abs(a(n)) > 2 is for n = 210. - Alonso del Arte, Oct 05 2014

Links

• Eric M. Schmidt, Table of n, a(n) for n = 1..10000
• Paul Erdős, Carl Pomerance and András Sárközy, On locally repeated values of certain arithmetic functions. II, Acta Math. Hungar. 49 (1987), 251-259.

Crossrefs

Cf. A001221 (omega).

Cf. A053222: first differences of sigma(n) = A000203.

Cf. A076191: first differences of bigomega(n) = A001222.

Cf. A127440: first differences of mobius(n) = A008683.

Programs

Maple

with(numtheory): A248211:=n->nops(factorset(n+1))-nops(factorset(n)): seq(A248211(n), n=1..100);

Mathematica

Table[PrimeNu[n + 1] - PrimeNu[n], {n, 100}] (* Hurt *)
Differences[PrimeNu[Range[100]]] (* Alonso del Arte, Oct 04 2014 *)

PARI

a(n) = omega(n+1) - omega(n); \\ Michel Marcus, Dec 29 2022

Formula

a(n) = omega(n+1) - omega(n) = A001221(n+1) - A001221(n).

G.f.: (1 - x)*Sum_{k>=1} x^(prime(k)-1)/(1 - x^prime(k)). - Ilya Gutkovskiy, Mar 15 2017