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A230543 Numbers n that form a Pythagorean quadruple with n', n'' and sqrt(n^2 + n'^2 + n''^2), where n' and n'' are the first and the second arithmetic derivative of n.

%I A230543 #32 Feb 16 2025 08:33:20
%S A230543 512,1203,3456,6336,23328,42768,157464,249753,288684,400000,722718,
%T A230543 1062882,1948617,2700000,4950000,18225000,33412500,105413504,
%U A230543 123018750,225534375,312500000,408918816
%N A230543 Numbers n that form a Pythagorean quadruple with n', n'' and sqrt(n^2 + n'^2 + n''^2), where n' and n'' are the first and the second arithmetic derivative of n.
%C A230543 Tested up to n = 4.09*10^8.
%H A230543 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PythagoreanQuadruple.html">Pythagorean Quadruple</a>
%e A230543 If n = 6336 then n' = 23808, n'' = 103936 and sqrt(n^2 + n'^2 + n''^2) = 106816.
%p A230543 with(numtheory): P:= proc(q) local a1, a2, n, p;
%p A230543 for n from 2 to q do a1:=n*add(op(2,p)/op(1,p),p=ifactors(n)[2]);
%p A230543 a2:=a1*add(op(2,p)/op(1,p),p=ifactors(a1)[2]);
%p A230543 if type(sqrt(n^2+a1^2+a2^2),integer) then print(n);
%p A230543 fi; od; end: P(10^10);
%Y A230543 Cf. A003415, A009003, A009004, A068346, A210503, A211176.
%Y A230543 Cf. A096907-A096909 and A097263-A097266 for Pythagorean Quadruples.
%K A230543 nonn,more
%O A230543 1,1
%A A230543 _Paolo P. Lava_, Oct 25 2013
%E A230543 a(16)-a(18) from _Giovanni Resta_, Oct 25 2013
%E A230543 a(19) from _Ray Chandler_, Dec 22 2016
%E A230543 a(20) from _Ray Chandler_, Dec 31 2016
%E A230543 a(21) from _Ray Chandler_, Jan 05 2017
%E A230543 a(22) from _Ray Chandler_, Jan 09 2017