A127440 -id:A127440 - cp's OEIS frontend

A127442 Triangle read by rows: row n consists of 1 followed by initial n-1 terms of A127440.

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

Offset: 1

Gary W. Adamson, Jan 14 2007

Row sums give A008683.

			Triangle begins:
1
1 -2,
1 -2, 0,
1 -2, 0, 1,
1 -2, 0, 1, -1,
1 -2, 0, 1, -1, 2,
1 -2, 0, 1, -1, 2, -2,
1 -2, 0, 1, -1, 2, -2, 1,
1 -2, 0, 1, -1, 2, -2, 1, 0,
1 -2, 0, 1, -1, 2, -2, 1, 0, 1,
1 -2, 0, 1, -1, 2, -2, 1, 0, 1, -2,

Cf. A127440 (first differences of mu(n), A008683).

Cf. A008683.

One term corrected by T. D. Noe, Feb 08 2007

Edited, corrected and extended by N. J. A. Sloane, Mar 01 2007

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

Original entry on oeis.org

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

Wesley Ivan Hurt, Oct 04 2014

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

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.

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.

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

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

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

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

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A127442 Triangle read by rows: row n consists of 1 followed by initial n-1 terms of A127440.

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

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