A083849 a(n) is the largest prime of the form x^2 + 1 <= 2^n.

2, 2, 5, 5, 17, 37, 101, 197, 401, 677, 1601, 3137, 8101, 15877, 32401, 62501, 122501, 246017, 512657, 1020101, 2073601, 4137157, 8386817, 16695397, 33339077, 66977857, 133772357, 268304401, 536663557, 1073610757, 2146098277

Offset: 1

Author

Harry J. Smith, May 05 2003

Keywords

• nonn

Comments

It is conjectured that this sequence is increasing, but this has never been proved.

It is easily shown that all terms greater than 5 end in 1 or 7.

References

• G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, th. 17.
• P. Ribenboim, The Little Book of Big Primes. Springer-Verlag, 1991, p. 190.

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..1000
• Eric Weisstein's World of Mathematics, Landau's Problems.

Crossrefs

Cf. A005574, A002496, A083844, A083845, A083846, A083847, A083848.

Programs

PARI

a(n) = my(last = 2^n+1); while ((p = precprime(last-1)) && (! issquare(p-1)), last = p;); p \\ Michel Marcus, Jun 14 2013

PARI

a(n)=my(k=sqrtint(2^n-1)); while(!isprime(k^2+1), k--); k^2+1 \\ Charles R Greathouse IV, Nov 29 2013