A306908 Numbers k with exactly three distinct prime factors and such that phi(k) is a square.

60, 114, 126, 170, 204, 240, 273, 285, 315, 364, 370, 380, 438, 444, 456, 468, 504, 540, 680, 816, 825, 902, 960, 969, 978, 1010, 1026, 1071, 1095, 1100, 1134, 1212, 1258, 1292, 1358, 1456, 1460, 1480, 1500, 1520, 1729, 1746, 1752, 1776, 1824, 1836, 1872

Offset: 1

Author

Bernard Schott, Mar 16 2019

Keywords

• nonn

Comments

This sequence is the intersection of A033992 and A039770.

The integers with only one prime factor and whose totient is a square are in A002496 and A054755, the integers with two prime factors and whose totient is a square are in A324745, A324746 and A324747.

Examples

			1st family: 273 = 3 * 7 * 13 and phi(273) = 12^2.
2nd family: 816 = 2^4 * 3 * 17 and phi(816) = 16^2.
3rd family: 6975 = 3^2 * 5^2 * 31 and phi(6975) = 60^2.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Bernard Schott, Subfamilies and subsequences

Crossrefs

Intersection of A033992 and A039770.

Cf. A002496, A054755 (only one prime factor), A324745, A324746, A324747 (two prime factors).

Programs

Maple

filter:= n -> issqr(numtheory:-phi(n)) and nops
(numtheory:-factorset(n))=3:
select(filter, [$1..2000]); # after Robert Israel in A324745

Mathematica

Select[Range[2000], And[PrimeNu@ # == 3, IntegerQ@ Sqrt@ EulerPhi@ #] &] (* Michael De Vlieger, Mar 31 2019 *)

PARI

isok(n) = (omega(n)==3) && issquare(eulerphi(n)); \\ Michel Marcus, Mar 19 2019

Formula

1st family: The primitive terms are p*q*r with p,q,r primes and phi(p*q*r) = (p-1)*(q-1)*(r-1) = m^2. These primitives generate the entire family formed by the numbers k = p^(2s+1) * q^(2t+1) * r^(2u+1) with s,t,u >= 0, and phi(k) = (p^s * q^t * r^u * m)^2.

2nd family: The primitive terms are p^2 * q * r with p,q,r primes and phi(p^2 * q * r) = p*(p-1)*(q-1)*(r-1) = m^2. These primitives generate the entire family formed by the numbers k = p^(2s) * q^(2t+1) * r^(2u+1) with s >= 1, t,u >= 0, and phi(k) = (p^(s-1) * q^t * r^u * m)^2.

3rd family: The primitive terms are p^2 * q^2 * r with p,q,r primes and phi(p^2 * q^2 * r) = p*q*(p-1)*(q-1)*(r-1) = m^2. These primitives generate the entire family formed by the numbers k = p^(2s) * q^(2t) * r^(2u+1) with s,t> = 1, u >= 0, and phi(k) = (p^(s-1) * q^(t-1) * r^u * m)^2.