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.

512, 1203, 3456, 6336, 23328, 42768, 157464, 249753, 288684, 400000, 722718, 1062882, 1948617, 2700000, 4950000, 18225000, 33412500, 105413504, 123018750, 225534375, 312500000, 408918816

Offset: 1

Author

Paolo P. Lava, Oct 25 2013

Keywords

• nonn
• more

Comments

Tested up to n = 4.09*10^8.

Examples

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

Links

• Eric Weisstein's World of Mathematics, Pythagorean Quadruple

Crossrefs

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

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

Programs

Maple

with(numtheory): P:= proc(q) local a1, a2, n, p;
for n from 2 to q do a1:=n*add(op(2,p)/op(1,p),p=ifactors(n)[2]);
a2:=a1*add(op(2,p)/op(1,p),p=ifactors(a1)[2]);
if type(sqrt(n^2+a1^2+a2^2),integer) then print(n);
fi; od; end: P(10^10);

Extensions

a(16)-a(18) from Giovanni Resta, Oct 25 2013

a(19) from Ray Chandler, Dec 22 2016

a(20) from Ray Chandler, Dec 31 2016

a(21) from Ray Chandler, Jan 05 2017

a(22) from Ray Chandler, Jan 09 2017