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EXAMPLE
| a(1)=1 because there is one solution (a,b,c) as (4,5,6) with 0<c<=10^1.
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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]
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EXTENSIONS
| First few terms found by Tito Piezas III (tpiezas(AT)gmail.com), 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(6) corrected and a(7) added by Max Alekseyev (maxale(AT)gmail.com), Jul 04 2011
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