A088165 NSW primes: NSW numbers that are also prime.

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

Offset: 1

Author

Christian Schroeder, Sep 21 2003

Keywords

• nonn

Comments

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

References

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

Links

• 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

Crossrefs

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

Programs

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

Formula

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

Extensions

More terms from Ray Chandler, Sep 21 2003