A084216 a(n) = smallest integer d such that a quadratic representation 2n+1 = x^2 + d*y^2 exists (x,y positive integers).

2, 1, 3, 2, 2, 1, 6, 1, 2, 3, 7, 1, 2, 1, 3, 2, 10, 1, 3, 1, 2, 1, 11, 3, 2, 1, 6, 2, 2, 1, 3, 1, 2, 5, 7, 1, 2, 7, 3, 2, 2, 1, 6, 1, 3, 3, 14, 1, 2, 1, 3, 5, 2, 1, 3, 1, 10, 1, 19, 2, 2, 1, 3, 2, 2, 3, 6, 1, 2, 5, 22, 1, 2, 1, 3, 1, 11, 1, 6, 5, 2, 11, 23, 1

Offset

1,1

Links

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Example

a(7)=6 because d=6 is the smallest d giving integer solutions x=2, y=1 in (2*7+1) = x^2+d*y^2 = 3^2 + 6*1^2 = 15.

Mathematica

Table[d = 1; While[! Resolve[Exists[{x, y}, And[2*n + 1 == x^2 + d*y^2, x > 0, y > 0]], Integers], d++]; d, {n, 120}] (* Michael De Vlieger, Mar 19 2026 *)

Crossrefs

Sequence in context: A385926 A162348 A262324 * A286364 A347240 A308751

Adjacent sequences: A084213 A084214 A084215 * A084217 A084218 A084219

Keyword

nonn

Author

Wouter Meeussen, Jun 20 2003

Extensions

Data corrected by Sean A. Irvine, Mar 19 2026

Status

approved