A121085 - OEIS

%I #12 Oct 17 2015 09:22:51

%S 3,30,293,2881,28871,288685,2886366,28868362,288673693,2886752763

%N Number of primitive Pythagorean-like triples a^2+b^2=c^2+k for k=-3 with 0<c<=10^n.

%C It is conjectured by the first author that a(n)/10^n as n->inf is 1/(2*sqrt(3)) = 0.28867...

%e a(1)=3 because there are 3 solutions (a,b,c) as (2,3,4),(5,6,8),(4,9,10) with 0<c<=10^1.

%t 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] (* Courtesy of Daniel Lichtblau of Wolfram Research *)

%Y Cf. A101931, A121082, A121083, A121084, A121086, A121087, A121088

%K more,nonn

%O 1,1

%A _Tito Piezas III_, Aug 11 2006

%E 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).

%E a(7) from _Max Alekseyev_, Jul 03 2011

%E a(8)-a(10) from _Hiroaki Yamanouchi_, Oct 17 2015