A308664 Numbers k such that tau(k) and phi(k) are the legs of a Pythagorean triple.

20, 36, 60, 100, 300

Offset

1,1

Comments

The sequence is finite since for all large enough n, we have tau(n) < n^(1/4) and phi(n) > n^(3/4) while, if x < y are the legs of a Pythagorean triangle, we always have y < x^2/2. - Giovanni Resta, Jul 27 2019

From Resta's inequality it can be deduced that phi(n) <= 2304. Then it's easy to see that the sequence is full. - Max Alekseyev, May 30 2024

Links

Table of n, a(n) for n=1..5.

Example

60 is in this sequence because tau(60) = 12 and phi(60) = 16, legs of the Pythagorean triple {12, 16, 20} (12^2 + 16^2 = 20^2).

Mathematica

Select[Range[300], IntegerQ@Sqrt[DivisorSigma[0, #]^2 + EulerPhi[#]^2] &] (* Amiram Eldar, Jul 26 2019 *)

Prog

(PARI) for(i = 1,  2000, a = eulerphi(i); b = numdiv(i); if(issquare(a^2 + b^2), print1(i, ", ")))

Crossrefs

Cf. A000005, A000010, A020488, A062516, A063469, A063470, A112954.

Sequence in context: A078210 A174628 A316098 * A184065 A259734 A294736

Adjacent sequences: A308661 A308662 A308663 * A308665 A308666 A308667

Keyword

nonn,fini,full

Author

Antonio Roldán, Jul 14 2019

Extensions

"full" keyword added by Max Alekseyev, May 30 2024

Status

approved