A005850 -id:A005850 - cp's OEIS frontend

A088165 NSW primes: NSW numbers that are also prime.

7, 41, 239, 9369319, 63018038201, 489133282872437279, 19175002942688032928599, 123426017006182806728593424683999798008235734137469123231828679

Offset: 1

Christian Schroeder, Sep 21 2003

nonn

Next term a(9) is too large (99 digits) to include here. - Ray Chandler, Sep 21 2003
These primes are the prime RMS numbers (A140480): primes p such that (1+p^2)/2 is a square r^2. Then r is a Pell number, A000129. - T. D. Noe, Jul 01 2008
Also prime numerators with an odd index in A001333. - Ctibor O. Zizka, Aug 13 2008
r in the above note of T. D. Noe is a prime Pell number (A000129) with an odd index. - Ctibor O. Zizka, Aug 13 2008
General recurrence is a(n) = (a(1)-1)*a(n-1) - a(n-2), a(1) >= 4, lim_{n->infinity} a(n) = x*(k*x+1)^n, k = a(1)-3, x = (1+sqrt((a(1)+1)/(a(1)-3)))/2. Examples in the OEIS: a(1)=4 gives A002878, primes in it A121534. a(1)=5 gives A001834, primes in it A086386. a(1)=6 gives A030221, primes in it not in the OEIS {29, 139, 3191, ...}. a(1)=7 gives A002315, primes in it A088165. a(1)=8 gives A033890, primes in it not in the OEIS (do there exist any ?). a(1)=9 gives A057080, primes in it not in the OEIS {71, 34649, 16908641, ...}. a(1)=10 gives A057081, primes in it not in the OEIS {389806471, 192097408520951, ...}. - Ctibor O. Zizka, Sep 02 2008

Paulo Ribenboim, The New Book of Prime Number Records, 3rd edition, Springer-Verlag, New York, 1995, pp. 367-369.

Alois P. Heinz, Table of n, a(n) for n = 1..14
Morris Newman, Daniel Shanks, H. C. Williams, Simple groups of square order and an interesting sequence of primes, Acta Arith., 38 (1980/1981), pp. 129-140.
M. Newman, D. Shanks and L. L. Foster, Simple groups of square order (6176), The American Mathematical Monthly, Vol. 86, No. 4 (Apr., 1979), pp. 314-315.
The Prime Glossary, NSW numbers

Cf. A002315 (NSW numbers), A005850 (indices for NSW primes).

w=3+quadgen(32); forprime(p=2,1e3, if(ispseudoprime(t=imag((1+w)*w^p)), print1(t", "))) \\ Charles R Greathouse IV, Apr 29 2015

a(n) mod A005850(n) = 1. - Altug Alkan, Mar 17 2016

More terms from Ray Chandler, Sep 21 2003

A099088 Indices of prime companion Pell numbers, divided by 2 (A001333).

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 16, 19, 29, 47, 59, 163, 257, 421, 937, 947, 1493, 1901, 6689, 8087, 9679, 28753, 79043, 129127, 145969, 165799, 168677, 170413, 172243, 278321, 552283

Offset: 1

T. D. Noe, Sep 24 2004

Note that for A001333(n) to be prime, the index n must be prime or a power of 2. The indices greater than 421 yield probable primes.

Numbers n for which ((1+sqrt(2))^n + (1-sqrt(2))^n)/2 is prime. - Artur Jasinski, Dec 10 2006

F. Le Lionnais, Les Nombres Remarquables. Paris: Hermann, p. 62, 1983.

J. B. Cosgrave and K. Dilcher, Pairs of reciprocal quadratic congruences involving primes, Fib. Quart. 51 (2) (2013) 98, after Theorem 3.
Eric Weisstein's World of Mathematics, Pell Number
Eric Weisstein's World of Mathematics, Integer Sequence Primes

Cf. A002203 (companion Pell numbers), A086395 (primes in A001333), A096650 (indices of prime Pell numbers).

Cf. A005850.

lst={}; a=1; b=1; Do[c=a+2b; a=b; b=c; If[PrimeQ[c], AppendTo[lst, n]], {n, 2, 10000}]; lst
(* Second program: *)
Do[If[PrimeQ[Expand[((1 + Sqrt[2])^n + (1 - Sqrt[2])^n)/2]], Print[n]], {n, 0, 1000}] (* Artur Jasinski, Dec 10 2006 *)

isok(n) = isprime(polchebyshev(n, 1, I)/I^n); \\ Michel Marcus, Dec 07 2018

a(24) from Eric W. Weisstein, May 22 2006
a(25) from Eric W. Weisstein, Aug 29 2006
a(26) from Eric W. Weisstein, Nov 11 2006
a(27) from Eric W. Weisstein, Nov 26 2006
a(28) from Eric W. Weisstein, Dec 10 2006
a(29) from Eric W. Weisstein, Jan 25 2007
a(30) from Robert Price, Dec 07 2018
a(31) from Robert Price, Dec 05 2023

cp's OEIS Frontend

A088165 NSW primes: NSW numbers that are also prime.

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