A083848 a(n)^2 + 1 is largest prime of the form x^2 + 1 <= 2^n.

1, 1, 2, 2, 4, 6, 10, 14, 20, 26, 40, 56, 90, 126, 180, 250, 350, 496, 716, 1010, 1440, 2034, 2896, 4086, 5774, 8184, 11566, 16380, 23166, 32766, 46326, 65534, 92666, 131070, 185354, 262130, 370714, 524260, 741454, 1048554, 1482904, 2097146

Offset

1,3

Comments

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

Ratio of successive terms appears to approach sqrt(2). - 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

Table of n, a(n) for n=1..42.

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

Mathematica

Join[{1}, Table[Module[{p=NextPrime[2^n, -1]}, While[!IntegerQ[Sqrt[p-1]], p=NextPrime[p, -1]]; Sqrt[p-1]], {n, 2, 30}]] (* The program generates the first 30 terms of the sequence. *) (* Harvey P. Dale, Jan 05 2026 *)

Crossrefs

Cf. A005574, A002496, A083844, A083845, A083846, A083847, A083849.

Sequence in context: A034406 A098330 A240310 * A330644 A278297 A139582

Adjacent sequences: A083845 A083846 A083847 * A083849 A083850 A083851

Keyword

nonn

Author

Harry J. Smith, May 05 2003

Status

approved