A121084
Number of primitive Pythagorean-like triples a^2+b^2=c^2+k for k=2 with 0
1, 10, 100, 983, 9912, 99211, 991714, 9918739, 99187754, 991897081
Offset: 1
Keywords
Examples
a(1)=1 because there is one solution (a,b,c) as (3,3,4) with 0<c<=10^1.
Crossrefs
Cf. A101931.
Programs
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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]
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(10) from Asif Ahmed, Dec 07 2024