%I A088165 #44 Nov 05 2025 15:35:31
%S A088165 7,41,239,9369319,63018038201,489133282872437279,
%T A088165 19175002942688032928599,
%U A088165 123426017006182806728593424683999798008235734137469123231828679
%N A088165 NSW primes: NSW numbers that are also prime.
%C A088165 Next term a(9) is too large (99 digits) to include here. - _Ray Chandler_, Sep 21 2003
%C A088165 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
%C A088165 Also prime numerators with an odd index in A001333. - _Ctibor O. Zizka_, Aug 13 2008
%C A088165 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
%C A088165 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
%D A088165 Paulo Ribenboim, The New Book of Prime Number Records, 3rd edition, Springer-Verlag, New York, 1995, pp. 367-369.
%H A088165 Alois P. Heinz, <a href="/A088165/b088165.txt">Table of n, a(n) for n = 1..14</a>
%H A088165 Morris Newman, Daniel Shanks, H. C. Williams, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa38/aa3826.pdf">Simple groups of square order and an interesting sequence of primes</a>, Acta Arith., 38 (1980/1981), pp. 129-140.
%H A088165 M. Newman, D. Shanks and L. L. Foster, <a href="https://www.jstor.org/stable/2320772">Simple groups of square order (6176)</a>, The American Mathematical Monthly, Vol. 86, No. 4 (Apr., 1979), pp. 314-315.
%H A088165 The Prime Glossary, <a href="https://t5k.org/glossary/page.php?sort=NSWNumber">NSW numbers</a>
%F A088165 a(n) mod A005850(n) = 1. - _Altug Alkan_, Mar 17 2016
%o A088165 (PARI) 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
%Y A088165 Cf. A002315 (NSW numbers), A005850 (indices for NSW primes).
%K A088165 nonn
%O A088165 1,1
%A A088165 _Christian Schroeder_, Sep 21 2003
%E A088165 More terms from _Ray Chandler_, Sep 21 2003