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

1,2

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)

Links

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

Example

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

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A119630 A216480 A257146 * A259946 A249911 A292223

Adjacent sequences: A291546 A291547 A291548 * A291550 A291551 A291552

Keyword

nonn,easy

Author

Amiram Eldar and Altug Alkan, Aug 26 2017

Status

approved