A128428 Number of distinct prime factors of n^2+1.

1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 3, 1, 2, 1, 3, 2, 2, 1, 3, 2, 3, 1, 2, 1, 3, 2, 2, 2, 3, 2, 3, 2, 2, 1, 3, 2, 2, 1, 2, 2, 3, 2, 2, 2, 4, 2, 2, 2, 2, 2, 3, 1, 3, 1, 3, 2, 2, 2, 2, 2, 3, 2, 2, 1, 3, 2, 2, 2, 2, 3, 4, 1, 3, 2, 3, 2, 2, 2, 3, 2, 4, 1, 2, 2, 3, 2, 3, 1, 3, 2, 3, 1, 2, 2, 3, 3, 3, 2, 2, 2

Offset: 1

Author

Kent Horvath (kenthorvath(AT)gmail.com), May 10 2007

Keywords

• nonn

Comments

a(n) is also the number of distinct prime factors, up to multiplication by units, of n + i in the ring of Gaussian integers. - Jason Kimberley, Dec 17 2011

Examples

			a(3) = 2 because 3^2+1 = 2*5.
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A193330 (counted with multiplicity).

Programs

Maple

a:= n-> nops(select(isprime, numtheory[divisors](n^2+1))):
seq(a(n), n=1..100);  # Alois P. Heinz, Dec 06 2020

Mathematica

a[n_]:=Length[FactorInteger[n^2 + 1]]

PARI

a(n)=omega(n^2+1) \\ Charles R Greathouse IV, Jul 31 2011

Formula

a(n) = A001221(n^2+1).