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A291549
Numbers n such that both phi(n) and psi(n) are perfect squares.
2
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
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
KEYWORD
nonn,easy
AUTHOR
Amiram Eldar and Altug Alkan, Aug 26 2017
STATUS
approved