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A083849
a(n) is the largest prime of the form x^2 + 1 <= 2^n.
7
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
KEYWORD
nonn
AUTHOR
Harry J. Smith, May 05 2003
STATUS
approved