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.

20, 100, 260, 340, 500, 740, 820, 1700, 2900, 3380, 4100, 5300, 5780, 6500, 7540, 8500, 8900, 9620, 9860, 10100, 11300, 12580, 13700, 13780, 13940, 14900, 15860, 16820, 17300, 18020, 18500, 18980, 19300, 19700, 22100, 23780, 25220, 27380, 28340, 29380, 30260, 30340, 30500, 30740, 33620, 34340, 35380, 35620, 36500, 37060

Offset: 1

Author

Zhi-Wei Sun, Oct 22 2017

Keywords

• nonn

Comments

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.

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.

Examples

			a(1) = 20 since 20^2 = 12^2 + 16^2 with 12, 16, 20 all practical and gcd(12,16,20) = 4.
a(2) = 100 since 100^2 = 28^2 + 96^2 with 28, 96, 100 all practical and gcd(28,96,100) = 4.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..80
• Zhi-Wei Sun, Conjectures on representations involving primes, 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.
• Li-Yuan Wang and Zhi-Wei Sun, On practical numbers of some special forms, arXiv:1809.01532 [math.NT], 2018.

Crossrefs

Cf. A000290, A005153, A156682.

Programs

Mathematica

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
f[n_]:=f[n]=FactorInteger[n];
Pow[n_, i_]:=Pow[n,i]=Part[Part[f[n], i], 1]^(Part[Part[f[n], i], 2]);
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}];
pr[n_]:=pr[n]=n>0&&(n<3||Mod[n, 2]+Con[n]==0);
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}]