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A294112 Practical numbers z such that z^2 = x^2 + y^2 for some practical numbers x and y with gcd(x,y,z) = 4.

%I A294112 #20 Jun 18 2025 03:18:55
%S A294112 20,100,260,340,500,740,820,1700,2900,3380,4100,5300,5780,6500,7540,
%T A294112 8500,8900,9620,9860,10100,11300,12580,13700,13780,13940,14900,15860,
%U A294112 16820,17300,18020,18500,18980,19300,19700,22100,23780,25220,27380,28340,29380,30260,30340,30500,30740,33620,34340,35380,35620,36500,37060
%N A294112 Practical numbers z such that z^2 = x^2 + y^2 for some practical numbers x and y with gcd(x,y,z) = 4.
%C A294112 Conjecture: The sequence has infinitely many terms. Also, there are infinitely many Pythagorean triples (x,y,z) with x,y,z all practical and gcd(x,y,z) = 6.
%C A294112 It is easy to show that there are no Pythagorean triples (x,y,z) with x,y,z all practical and gcd(x,y,z) = 2.
%H A294112 Zhi-Wei Sun, <a href="/A294112/b294112.txt">Table of n, a(n) for n = 1..80</a>
%H A294112 Zhi-Wei Sun, <a href="http://arxiv.org/abs/1211.1588">Conjectures on representations involving primes</a>, in: M. B. Nathanson (ed.), Combinatorial and Additive Number Theory II: CANT, New York, NY, USA, 2015 and 2016, Springer Proc. in Math. & Stat., Vol. 220, Springer, New York, 2017; arXiv:1211.1588 [math.NT], 2012-2017.
%H A294112 Li-Yuan Wang and Zhi-Wei Sun, <a href="https://arxiv.org/abs/1809.01532">On practical numbers of some special forms</a>, arXiv:1809.01532 [math.NT], 2018.
%e A294112 a(1) = 20 since 20^2 = 12^2 + 16^2 with 12, 16, 20 all practical and gcd(12,16,20) = 4.
%e A294112 a(2) = 100 since 100^2 = 28^2 + 96^2 with 28, 96, 100 all practical and gcd(28,96,100) = 4.
%t A294112 SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
%t A294112 f[n_]:=f[n]=FactorInteger[n];
%t A294112 Pow[n_, i_]:=Pow[n,i]=Part[Part[f[n], i], 1]^(Part[Part[f[n], i], 2]);
%t A294112 Con[n_]:=Con[n]=Sum[If[Part[Part[f[n], s+1], 1]<=DivisorSigma[1, Product[Pow[n, i], {i, 1, s}]]+1, 0, 1], {s, 1, Length[f[n]]-1}];
%t A294112 pr[n_]:=pr[n]=n>0&&(n<3||Mod[n, 2]+Con[n]==0);
%t A294112 n=0;Do[If[pr[4m]==False,Goto[aa]];Do[If[SQ[(4m)^2-x^2]&&GCD[x,4m,Sqrt[(4m)^2-x^2]]==4&&pr[x]&&pr[Sqrt[(4m)^2-x^2]],n=n+1;Print[n," ",4m];Goto[aa]],{x,1,Sqrt[8]m}];Label[aa],{m,1,9265}]
%Y A294112 Cf. A000290, A005153, A156682.
%K A294112 nonn
%O A294112 1,1
%A A294112 _Zhi-Wei Sun_, Oct 22 2017