login
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.
5

%I #31 Jan 09 2017 22:45:29

%S 512,1203,3456,6336,23328,42768,157464,249753,288684,400000,722718,

%T 1062882,1948617,2700000,4950000,18225000,33412500,105413504,

%U 123018750,225534375,312500000,408918816

%N 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 Tested up to n = 4.09*10^8.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PythagoreanQuadruple.html">Pythagorean Quadruple</a>

%e If n = 6336 then n' = 23808, n'' = 103936 and sqrt(n^2 + n'^2 + n''^2) = 106816.

%p with(numtheory): P:= proc(q) local a1, a2, n, p;

%p for n from 2 to q do a1:=n*add(op(2,p)/op(1,p),p=ifactors(n)[2]);

%p a2:=a1*add(op(2,p)/op(1,p),p=ifactors(a1)[2]);

%p if type(sqrt(n^2+a1^2+a2^2),integer) then print(n);

%p fi; od; end: P(10^10);

%Y Cf. A003415, A009003, A009004, A068346, A210503, A211176.

%Y Cf. A096907-A096909 and A097263-A097266 for Pythagorean Quadruples.

%K nonn,more

%O 1,1

%A _Paolo P. Lava_, Oct 25 2013

%E a(16)-a(18) from _Giovanni Resta_, Oct 25 2013

%E a(19) from _Ray Chandler_, Dec 22 2016

%E a(20) from _Ray Chandler_, Dec 31 2016

%E a(21) from _Ray Chandler_, Jan 05 2017

%E a(22) from _Ray Chandler_, Jan 09 2017