A121087 Number of primitive Pythagorean-like triples a^2+b^2=c^2+k for k=-5 with 0<c<=10^n.

1, 22, 223, 2217, 22354, 223667, 2235713, 22360389, 223610157

Offset

1,2

Comments

It is conjectured by the first author that a(n)/10^n as n->inf is 1/(2*sqrt(5)) = 0.22360...

Links

Table of n, a(n) for n=1..9.

Example

a(1)=1 because there is one solution (a,b,c) as (2,4,5) with 0<c<=10^1.

Mathematica

(* Courtesy of Daniel Lichtblau of Wolfram Research *)
countTriples[m_, k_] := Module[ {c2, c2odd, total = 0, fax, g}, Do[ c2 = c^2 + k; If[c2 < 2, Continue[]]; c2odd = c2; While[EvenQ[c2odd], c2odd /= 2]; If [c2odd==1, If [OddQ[Log[2, c2]], total++ ]; Continue[]]; If[Mod[c2odd, 4] == 3, Continue[]]; g = GCD[c2odd, 100947]; If[g != 1 && g^2 != GCD[c2odd, 10190296809], Continue[]]; fax = Map[{Mod[ #[[1]], 4], #[[2]]}&, FactorInteger[c2odd]]; If[Apply[Or, Map[ #[[1]] == 3 && OddQ[ #[[2]]] &, fax]], Continue []]; fax = Cases[fax, {1, aa_}:>aa+1]; fax = Ceiling[Apply[Times, fax]/2]; total += fax; , {c, m}]; total]

Crossrefs

Cf. A101931.

Sequence in context: A331882 A008453 A290362 * A133719 A008948 A003909

Adjacent sequences: A121084 A121085 A121086 * A121088 A121089 A121090

Keyword

nonn

Author

Tito Piezas III, Aug 11 2006

Extensions

First few terms found by Tito Piezas III, James Waldby (j-waldby(AT)pat7.com)

Subsequent terms found by Andrzej Kozlowski (akoz(AT)mimuw.edu.pl), Daniel Lichtblau (danl(AT)wolfram.com)

a(7) from Max Alekseyev, May 30 2007

a(8)-a(9) from Lars Blomberg, Dec 22 2015

Status

approved