A002335 Least positive integer y such that A038873(n) = x^2 - 2y^2 for some x. (Formerly M0139 N0055)

1, 1, 2, 1, 3, 2, 1, 5, 2, 1, 4, 6, 3, 2, 7, 4, 3, 1, 7, 4, 9, 1, 8, 5, 10, 4, 7, 3, 2, 5, 8, 12, 2, 1, 9, 11, 8, 4, 7, 2, 1, 14, 6, 9, 5, 11, 13, 2, 14, 16, 4, 11, 8, 3, 2, 7, 10, 17, 12, 11, 1, 7, 13, 10, 6, 4, 3, 1, 16, 7, 20, 13, 5, 15, 4, 12, 2, 21, 14, 11, 7, 16, 13, 18, 5, 20, 9, 1, 8, 17, 14

Offset

1,3

Comments

A prime p is representable in the form x^2-2y^2 iff p is 2 or p == 1 or 7 mod 8. - Pab Ter (pabrlos2(AT)yahoo.com), Oct 22 2005

References

A. J. C. Cunningham, Quadratic Partitions. Hodgson, London, 1904, p. 1.

D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, p. 55.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Robert Israel, Table of n, a(n) for n = 1..10000

A. J. C. Cunningham, Quadratic Partitions, Hodgson, London, 1904 [Annotated scans of selected pages]

Maple

with(numtheory): readlib(issqr):for i from 1 to 300 do p:=ithprime(i): pmod8:=modp(p, 8): if p=2 or pmod8=1 or pmod8=7 then for y from 1 do if issqr(p+2*y^2) then printf("%d, ", y): break fi od fi od: # Pab Ter, Oct 22 2005

Mathematica

maxPrimePi = 200;
Reap[Do[If[MatchQ[Mod[p, 8], 1|2|7], rp = Reduce[x > 0 && y > 0 && p == x^2 - 2*y^2, {x, y}, Integers]; If[rp =!= False, xy = {x, y} /. {ToRules[rp /. C[1] -> 1]}; y0 = xy[[All, 2]] // Min // Simplify; Print[{p, xy[[1]]} ]; Sow[y0]]], {p, Prime[Range[maxPrimePi]]}]][[2, 1]] (* Jean-François Alcover, Oct 27 2019 *)

Crossrefs

Cf. A002334, A035251.

Sequence in context: A209073 A332077 A220901 * A280738 A370628 A207375

Adjacent sequences: A002332 A002333 A002334 * A002336 A002337 A002338

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

More terms from Pab Ter (pabrlos2(AT)yahoo.com), Oct 22 2005

Status

approved