A216784 a(n) is the number of distinct prime divisors of n^2 + 1 of the form m^2 + 1.

1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 0, 1, 1, 2, 2, 1, 1, 1, 1, 3, 0, 1, 0, 3, 1, 1, 0, 1, 1, 2, 1, 2, 1, 2, 1, 1, 0, 1, 1, 2, 1, 1, 1, 2, 2, 1, 0, 1, 2, 2, 1, 1, 0, 2, 1, 1, 1, 2, 1, 2, 1, 1, 0, 2

Offset: 1

Author

Michel Lagneau, Oct 15 2012

Keywords

• nonn

Comments

a(m) = 0 for m = A217276(n).

Examples

			a(13) = 3 because 13^2+1 = 170 = 2*5*17 with 3 divisors of the form m^2+1 such that 2 = 1^2+1, 5=2^2+1 and 17 = 4^2+1.
a(34) = 0 because 34^2+1 = 1157 = 13*89 and the prime divisors 13, 89 are not of the form m^2+1.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A217276, A181436, A180252.

Programs

Maple

with(numtheory):for n from 1 to 100 do:x:=n^2+1:y:=factorset(x):n1:=nops(y):i:=0:for k from 1 to n1 do:z:=sqrt(y[k]-1):if z=floor(z) then i:=i+1:else fi:od: printf(`%d, `,i):od:
# second Maple program:
a:= n-> nops(select(x-> issqr(x-1), ifactors(n^2+1)[2][..., 1])):
seq(a(n), n=1..87);  # Alois P. Heinz, Jul 24 2025

Mathematica

a[n_] := Length @ Select[FactorInteger[n^2 + 1][[;;,1]], IntegerQ @ Sqrt[# - 1] &]; Array[a, 100] (* Amiram Eldar, Sep 11 2019 *)