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Numbers n such that (n, sigma(n), tau(n)) lies on a sphere with integral radius centered at the origin, i.e., n^2 + sigma(n)^2 + tau(n)^2 is a square.
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%I #15 Jul 17 2019 04:20:03

%S 6,61,2089,3606,18585,28710,70981,121374,176529,520320,970783,

%T 62788800,114682878,144653952,174635334,182054895,3857228169,

%U 4349012190,15994971740,17587660602,22842677823,65183331200,66928439760

%N Numbers n such that (n, sigma(n), tau(n)) lies on a sphere with integral radius centered at the origin, i.e., n^2 + sigma(n)^2 + tau(n)^2 is a square.

%C a(17) > 3*10^9. - _Jinyuan Wang_, Jul 09 2019

%C a(24) > 10^12, if it exists. - _Giovanni Resta_, Jul 17 2019

%e 6^2 + tau(6)^2 + sigma(6)^2 = 36 + 16 + 144 = 196 = 14^2. So 6 is in the sequence.

%o (PARI) isok(n) = issquare(n^2 + numdiv(n)^2 + sigma(n)^2);

%Y Cf. A000005 (tau), A000203 (sigma), A066764 (analog in 2d).

%K nonn,more

%O 1,1

%A _Michel Marcus_, Jun 05 2014

%E a(13)-a(16) from _Jinyuan Wang_, Jul 09 2019

%E a(17)-a(23) from _Giovanni Resta_, Jul 17 2019