A087952 - OEIS

A087952

Smallest prime == 1 (mod n) and > n^2.

2, 5, 13, 17, 31, 37, 71, 73, 109, 101, 199, 157, 313, 197, 241, 257, 307, 379, 419, 401, 463, 617, 599, 577, 701, 677, 757, 953, 929, 991, 1117, 1153, 1123, 1259, 1471, 1297, 1481, 1483, 1873, 1601, 1723, 1933, 1979, 2069, 2161, 2347, 2351, 2593, 2549, 2551

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OFFSET

1,1

COMMENTS

Primes arising in A087554.

Since A014085(n) ~ n/log(n) one may conjecture that a(n) < 2*n^2 for all n > 1. Numerically we find a(n) = n^2*(1 + O(1/sqrt(n))). - M. F. Hasler, Feb 27 2020

LINKS

M. F. Hasler, Table of n, a(n) for n = 1..10000, Feb 27 2020

EXAMPLE

For n=1, a(1) = 2, because 2 == 1 mod 1 and 2 > 1^2.

For n=2, a(2) = 5, because 5 == 1 mod 2 and 5 > 2^2.

MATHEMATICA

spr[n_]:=Module[{p=NextPrime[n^2]}, While[Mod[p, n]!=1, p=NextPrime[p]]; p]; Join[ {2}, Array[spr, 50, 2]] (* Harvey P. Dale, Jun 21 2021 *)

PROG

(PARI) apply( {A087952(n)=forprime(p=n^2+1, , (p-1)%n||return(p))}, [1..66]) \\ M. F. Hasler, Feb 27 2020

CROSSREFS

Cf. A034693, A034694, A087554.

Cf. A014085 (number of primes between n^2 and (n+1)^2).