A245199 Numbers n where phi(n) and tau(n) are perfect squares.

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

Robert Israel, Table of n, a(n) for n = 1..8000

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

Cf. A000005, A000010, A036436, A039770.

Sequence in context: A230110 A223587 A302312 * A303042 A242506 A302879

Adjacent sequences: A245196 A245197 A245198 * A245200 A245201 A245202

Keyword

nonn,easy

Author

Reinhard Muehlfeld, Jul 13 2014

Status

approved