NSW Number

The numbers

$$S_{2m+1}={(1+\sqrt{2}\,)^{2m+1}+(1-\sqrt{2}\,)^{2m+1}\over 2}$$

for positive integer $m$. The first few terms are 1, 7, 41, 239, 1393, ... (Sloane's A002315). The indices giving Prime NSW numbers are 3, 5, 7, 19, 29, 47, 59, 163, 257, 421, 937, 947, 1493, 1901, ... (Sloane's A005850).

References

Ribenboim, P. ``The NSW Primes.'' §5.9 in The New Book of Prime Number Records. New York: Springer-Verlag, pp. 367-369, 1996.

Sloane, N. J. A. Sequences A002315/M4423 and A005850/M2426 in ``An On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.