A291549 Numbers n such that both phi(n) and psi(n) are perfect squares.

1, 60, 170, 240, 315, 540, 679, 680, 960, 1500, 2142, 2160, 2720, 2835, 3840, 4250, 4365, 4860, 5770, 6000, 7875, 8568, 8640, 9154, 9809, 10880, 13500, 14322, 15360, 15435, 17000, 19278, 19440, 22413, 23080, 24000, 25515, 29682, 33271, 34272, 34560, 36616, 37114, 37500

Offset: 1

Author

Amiram Eldar and Altug Alkan, Aug 26 2017

Keywords

• nonn
• easy

Comments

Intersection of A039770 and A291167.

Squarefree terms are 1, 170, 679, 5770, 9154, 9809, 14322, ...

From Robert Israel, May 16 2019: (Start)

If n is in the sequence and p is a prime factor of n then p^2*n is in the sequence.

If n and m are coprime members of the sequence, then n*m is in the sequence. (End)

Examples

			60 is a term because phi(60) = 16 and psi(60) = 144 are both perfect squares.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..2000

Crossrefs

Cf. A000010, A000290, A001615, A039770, A291167.

Programs

Maple

filter:= proc(n) local F,psi,phi,p;
   F:= numtheory:-factorset(n);
   issqr( n*mul(1-1/p, p=F)) and issqr(n*mul(1+1/p,p=F))
end proc:
select(filter, [$1..50000]); # Robert Israel, May 15 2019

Mathematica

Select[Range[10^5], AllTrue[{EulerPhi@ #, If[# < 1, 0, # Sum[MoebiusMu[d]^2/d, {d, Divisors@ #}]]}, IntegerQ@ Sqrt@ # &] &] (* Michael De Vlieger, Aug 26 2017, after Michael Somos at A001615 *)

PARI

a001615(n) = my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1))
isok(n) = issquare(eulerphi(n)) && issquare(a001615(n)); \\ after Charles R Greathouse IV at A001615