A193432 Number of divisors of n^2 + 1.

1, 2, 2, 4, 2, 4, 2, 6, 4, 4, 2, 4, 4, 8, 2, 4, 2, 8, 6, 4, 2, 8, 4, 8, 2, 4, 2, 8, 4, 4, 4, 8, 6, 8, 4, 4, 2, 8, 6, 4, 2, 6, 4, 12, 4, 4, 4, 16, 4, 4, 4, 4, 4, 8, 2, 8, 2, 16, 4, 4, 4, 4, 4, 8, 4, 4, 2, 8, 8, 4, 6, 4, 8, 16, 2, 8, 4, 8, 4, 4, 4, 8, 6, 16, 2

Offset: 0

Author

Michel Lagneau, Jul 28 2011

Keywords

• nonn

Examples

			a(7) = 6 because 7^2 + 1 = 50 and the 6 divisors are {1, 2, 5, 10, 25, 50}.
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000
• Anton Shakov, Polynomials in Z[x] whose divisors are enumerated by SL_2(N_0), arXiv:2405.03552 [math.NT], 2024.

Crossrefs

Cf. A000005, A002522, A128428, A193330, A193433.

Programs

Maple

with(numtheory):for n from 0 to 110 do:n1:=nops(divisors(n^2+1)):s:=0:for m from 1 to n1 do: s:=s+1:od: printf(`%d, `, s):od:

Mathematica

Array[DivisorSigma[0, #^2 + 1] &, 85, 0] (* Michael De Vlieger, Mar 17 2018 *)

PARI

a(n) = numdiv(n^2+1); \\ Michel Marcus, Mar 16 2018

Python

from sympy import divisor_count
def A193432(n): return divisor_count(n**2+1) # Chai Wah Wu, Apr 17 2025

Formula

a(n) = A000005(A002522(n)). - Michel Marcus, Mar 16 2018