A089120 Smallest prime factor of n^2 + 1.

2, 5, 2, 17, 2, 37, 2, 5, 2, 101, 2, 5, 2, 197, 2, 257, 2, 5, 2, 401, 2, 5, 2, 577, 2, 677, 2, 5, 2, 17, 2, 5, 2, 13, 2, 1297, 2, 5, 2, 1601, 2, 5, 2, 13, 2, 29, 2, 5, 2, 41, 2, 5, 2, 2917, 2, 3137, 2, 5, 2, 13, 2, 5, 2, 17, 2, 4357, 2, 5, 2, 13, 2, 5, 2, 5477, 2, 53, 2, 5, 2, 37, 2, 5, 2

Offset: 1

Author

Cino Hilliard, Dec 05 2003

Keywords

• easy
• nonn

Comments

This includes A002496, primes that are of the form n^2+1.

Note that a(n) is the smallest prime p such that n^(p+1) == -1 (mod p). - Thomas Ordowski, Nov 08 2019

References

• H. Rademacher, Lectures on Elementary Number Theory, pp. 33-38.

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A002496, A014442, A085722, A144255, A193432.

Programs

Magma

[Min(PrimeDivisors(n^2+1)):n in [1..100]]; // Marius A. Burtea, Nov 13 2019

Mathematica

Array[FactorInteger[#^2 + 1][[1, 1]] &, {83}] (* Michael De Vlieger, Sep 08 2015 *)

PARI

smallasqp1(m) = { for(a=1,m, y=a^2 + 1; f = factor(y); v = component(f,1); v1 = v[length(v)]; print1(v[1]",") ) }

PARI

A089120(n)=factor(n^2+1)[1,1]  \\ M. F. Hasler, Mar 11 2012

Formula

a(2k+1)=2; a(10k +/- 2)=5, else a(26k +/- 8)=13, else a(34k +/- 4)=17, else a(58k +/- 12)=29, else a(74k +/- 6)=37,... - M. F. Hasler, Mar 11 2012

A089120(n) = 2 if n is odd, else A089120(n) = min { A002144(k) | n = +/- A209874(k) (mod 2*A002144(k)) }.