OFFSET
1,1
COMMENTS
It is conjectured by the first author that a(n)/10^n as n->inf is 1/(2*sqrt(3)) = 0.28867...
EXAMPLE
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.
MATHEMATICA
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 *)
KEYWORD
more,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, Jul 03 2011
a(8)-a(10) from Hiroaki Yamanouchi, Oct 17 2015
STATUS
approved