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%I M0139 N0055 #38 Oct 27 2019 14:05:30
%S 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,
%T 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,
%U 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
%N Least positive integer y such that A038873(n) = x^2 - 2y^2 for some x.
%C 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
%D A. J. C. Cunningham, Quadratic Partitions. Hodgson, London, 1904, p. 1.
%D D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, p. 55.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Robert Israel, <a href="/A002335/b002335.txt">Table of n, a(n) for n = 1..10000</a>
%H A. J. C. Cunningham, <a href="/A002330/a002330.pdf">Quadratic Partitions</a>, Hodgson, London, 1904 [Annotated scans of selected pages]
%p 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
%t maxPrimePi = 200;
%t 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 *)
%Y Cf. A002334, A035251.
%K nonn,easy
%O 1,3
%A _N. J. A. Sloane_
%E More terms from Pab Ter (pabrlos2(AT)yahoo.com), Oct 22 2005
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