A291549 - OEIS

%I #22 May 16 2019 08:45:02

%S 1,60,170,240,315,540,679,680,960,1500,2142,2160,2720,2835,3840,4250,

%T 4365,4860,5770,6000,7875,8568,8640,9154,9809,10880,13500,14322,15360,

%U 15435,17000,19278,19440,22413,23080,24000,25515,29682,33271,34272,34560,36616,37114,37500

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

%C Intersection of A039770 and A291167.

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

%C From _Robert Israel_, May 16 2019: (Start)

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

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

%H Robert Israel, <a href="/A291549/b291549.txt">Table of n, a(n) for n = 1..2000</a>

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

%p filter:= proc(n) local F,psi,phi,p;

%p    F:= numtheory:-factorset(n);

%p    issqr( n*mul(1-1/p, p=F)) and issqr(n*mul(1+1/p,p=F))

%p end proc:

%p select(filter, [$1..50000]); # _Robert Israel_, May 15 2019

%t 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 *)

%o (PARI) a001615(n) = my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1))

%o isok(n) = issquare(eulerphi(n)) && issquare(a001615(n)); \\ after _Charles R Greathouse IV_ at A001615

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

%K nonn,easy

%O 1,2

%A _Amiram Eldar_ and _Altug Alkan_, Aug 26 2017