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A245199
Numbers n where phi(n) and tau(n) are perfect squares.
1
1, 8, 10, 34, 57, 74, 85, 125, 185, 202, 219, 394, 451, 456, 489, 505, 514, 546, 570, 629, 640, 679, 680, 802, 985, 1000, 1026, 1057, 1154, 1285, 1354, 1365, 1387, 1417, 1480, 1717, 1752, 1938, 2005, 2016, 2047, 2176, 2190, 2340, 2457, 2509, 2565, 2594, 2649
OFFSET
1,2
COMMENTS
Numbers n such that A000005(n) and A000010(n) are perfect squares.
Intersection of A036436 and A039770. - Michel Marcus, Jul 15 2014
LINKS
EXAMPLE
8 is in the sequence because phi(8) = 4, tau(8) = 4, and 4 is a perfect square.
12 is not in the sequence because tau(12) = 6 is not a square.
MAPLE
filter:= proc(n) uses numtheory; issqr(phi(n)) and issqr(tau(n)) end proc:
select(filter, [$1..1000]); # Robert Israel, Jul 27 2014
MATHEMATICA
fQ[n_] := IntegerQ@ Sqrt@ EulerPhi[n] && IntegerQ@ Sqrt@ DivisorSigma[0, n]; Select[ Range@ 3000, fQ] (* Robert G. Wilson v, Jul 21 2014 *)
Select[Range[3000], AllTrue[{Sqrt[EulerPhi[#]], Sqrt[DivisorSigma[0, #]]}, IntegerQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Apr 01 2018 *)
PROG
(PARI) isok(n) = issquare(numdiv(n)) && issquare(eulerphi(n)); \\ Michel Marcus, Jul 15 2014
(Python)
from sympy import totient, divisor_count
from gmpy2 import is_square
[n for n in range(1, 10**4) if is_square(int(divisor_count(n))) and is_square(int(totient(n)))] # Chai Wah Wu, Aug 04 2014
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Reinhard Muehlfeld, Jul 13 2014
STATUS
approved