A094786 -id:A094786 - cp's OEIS frontend

A028871 Primes of the form k^2 - 2.

2, 7, 23, 47, 79, 167, 223, 359, 439, 727, 839, 1087, 1223, 1367, 1847, 2207, 2399, 3023, 3719, 3967, 4759, 5039, 5623, 5927, 7919, 8647, 10607, 11447, 13687, 14159, 14639, 16127, 17159, 18223, 19319, 21023, 24023, 25919, 28559, 29927

Offset: 1

Patrick De Geest

Except for the initial term, primes equal to the product of two consecutive even numbers minus 1. - Giovanni Teofilatto, Sep 24 2004
With exception of the first term 2, primes p such that continued fraction of (1+sqrt(p))/2 have period 4. - Artur Jasinski, Feb 03 2010
Subsequence of A094786. First primes q that are in A094786 but not here are 241, 3373, 6857, 19681, 29789, 50651, 300761, 371291, ...; q+2 are perfect powers m^k with odd k>1. - Zak Seidov, Dec 09 2014

D. Shanks, Solved and Unsolved Problems in Number Theory, 2nd. ed., Chelsea, 1978, p. 31.

Nathaniel Johnston, Table of n, a(n) for n = 1..10000
P. De Geest, Palindromic Quasipronics of the form n(n+x)
Eric Weisstein's World of Mathematics, Near-Square Prime
Wikipedia, Bunyakovsky conjecture

Cf. A028870, A089623, A010051, A094786; subsequence of A008865.

a028871 n = a028871_list !! (n-1)
a028871_list = filter ((== 1) . a010051') a008865_list
-- Reinhard Zumkeller, May 06 2013

[p: p in PrimesUpTo(100000)| IsSquare(p+2)]; // Vincenzo Librandi, Jun 19 2014

select(isprime, [2,seq((2*n+1)^2-2, n=1..1000)]); # Robert Israel, Dec 09 2014

lst={};Do[s=n^2;If[PrimeQ[p=s-2], AppendTo[lst, p]], {n, 6!}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 26 2008 *)
aa = {}; Do[If[4 == Length[ContinuedFraction[(1 + Sqrt[Prime[m]])/2][[2]]], AppendTo[aa, Prime[m]]], {m, 1, 1000}]; aa (* Artur Jasinski, Feb 03 2010 *)
Select[Table[n^2 - 2, {n, 400}], PrimeQ] (* Vincenzo Librandi, Jun 19 2014 *)

list(lim)=select(n->isprime(n),vector(sqrtint(floor(lim)+2),k,k^2-2)) \\ Charles R Greathouse IV, Jul 25 2011

a(n) = A028870(n)^2 -2. - R. J. Mathar, Dec 12 2023

A155897 Square matrix T(m,n)=1 if (2m+1)^n-2 is prime, 0 otherwise; read by antidiagonals.

Original entry on oeis.org

0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0

Offset: 1

M. F. Hasler, Feb 01 2009

In some sense a "minimal" possible generalization of the pattern of Mersenne primes (cf. A000043) is to consider powers of odd numbers (> 1) minus 2. Since even powers obviously correspond to an odd power of the base squared, it is sufficient to consider only odd powers, cf. A155899.

Cf. A084714, A128472, A094786, A014224, A109080, A090669, A128455, A128457, A128458, A128459, A128460, A128461.

T = matrix( 19,19,m,n, isprime((2*m+1)^n-2)) ;
A155897 = concat( vector( vecmin( matsize(T)), i, vector( i, j, T[j,i-j+1])))

cp's OEIS Frontend

A028871 Primes of the form k^2 - 2.

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