A083845 a(n)^2 + 1 is largest prime of the form x^2 + 1 <= 10^n.

2, 6, 26, 94, 314, 986, 3160, 9990, 31614, 99996, 316206, 999960, 3162246, 9999960, 31622764, 99999966, 316227734, 999999924, 3162277660, 9999999956, 31622776500, 99999999964, 316227766006, 999999999886, 3162277660140

Offset

1,1

Comments

It is conjectured that the number of primes of the form x^2+1 is infinite and thus this sequence never becomes a constant, but this has not been proved.

The ratio a(n+2)/a(n) appears to approach 10, as one might expect. - Bill McEachen, Nov 03 2013

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

Robert Israel, Table of n, a(n) for n = 1..1000

Eric Weisstein's World of Mathematics, Landau's Problems.

Maple

f:= proc(n) local t, t0;
if n::even then t0:= 10^(n/2)-1
else
  Digits:= n;
  t0:= floor(evalf(sqrt(10^n-1)));
  while (t0+1)^2+1 <= 10^n do t0:= t0+1 od:
fi;
for t from t0 by -1 do if isprime(t^2+1) then return t fi od
end proc:
map(f, [$1..50]); # Robert Israel, Jan 22 2026

Mathematica

Do[ k = Floor[ Sqrt[ 10^n] - 1]; While[ !PrimeQ[k^2 + 1], k-- ]; Print[k], {n, 1, 25}]

Prog

(PARI) a(n) = my(x=sqrtint(10^n-1)); while (!ispseudoprime(x^2+1), x--); x; \\ Michel Marcus, Jan 23 2026

Crossrefs

Cf. A005574, A002496, A083844, A083846, A083847, A083848, A083849.

Sequence in context: A027207 A027231 A393071 * A383933 A027239 A191821

Adjacent sequences: A083842 A083843 A083844 * A083846 A083847 A083848

Keyword

nonn

Author

Harry J. Smith, May 05 2003

Extensions

Edited and extended by Robert G. Wilson v, May 08 2003

a(19) corrected by Robert Israel, Jan 22 2026

Status

approved