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

1,1

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.

Prog

(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

Crossrefs

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

Sequence in context: A245849 A245850 A245844 * A326512 A325983 A063501

Adjacent sequences: A083846 A083847 A083848 * A083850 A083851 A083852

Keyword

nonn

Author

Harry J. Smith, May 05 2003

Status

approved