A035091 Smallest prime == 1 mod (n^2).

2, 5, 19, 17, 101, 37, 197, 193, 163, 101, 727, 433, 677, 197, 1801, 257, 3469, 1297, 10831, 401, 883, 1453, 12697, 577, 11251, 677, 1459, 3137, 10093, 1801, 15377, 12289, 2179, 3469, 7351, 1297, 5477, 18773, 9127, 1601, 16811, 3529, 22189, 11617

Offset: 1

Author

Labos Elemer

Keywords

• nonn

Comments

Smallest prime of form (n^2)*k+1, i.e., an arithmetic progression with n^2 differences; k is the subscript of the progressions.

Examples

			a(5) = 101 because in 5^2k + 1 = 25k + 1 progression k=4 generates the smallest prime (this is 101) and 26, 51, and 76 are composite.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..500 from Robert Price)
• Index entries for sequences related to primes in arithmetic progressions.

Crossrefs

Analogous case is A034694. Special case is A002496.

Cf. A070844 to A070858, A061092, A035095.

Programs

Mathematica

With[{prs=Prime[Range[2500]]},Flatten[Table[Select[prs,Mod[#-1,n^2]==0&,1],{n,50}]]] (* Harvey P. Dale, Sep 22 2021 *)

PARI

a(n) = if(n == 1, 2, my(s = n^2); forprime(p = 1, , if(p % s == 1, return(p)))); \\ Amiram Eldar, Mar 16 2025