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A306908 Numbers k with exactly three distinct prime factors and such that phi(k) is a square. 4

%I

%S 60,114,126,170,204,240,273,285,315,364,370,380,438,444,456,468,504,

%T 540,680,816,825,902,960,969,978,1010,1026,1071,1095,1100,1134,1212,

%U 1258,1292,1358,1456,1460,1480,1500,1520,1729,1746,1752,1776,1824,1836,1872

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

%C This sequence is the intersection of A033992 and A039770.

%C 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.

%H Robert Israel, <a href="/A306908/b306908.txt">Table of n, a(n) for n = 1..10000</a>

%H Bernard Schott, <a href="/A306908/a306908.pdf">Subfamilies and subsequences</a>

%F 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.

%F 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.

%F 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.

%e 1st family: 273 = 3 * 7 * 13 and phi(273) = 12^2.

%e 2nd family: 816 = 2^4 * 3 * 17 and phi(816) = 16^2.

%e 3rd family: 6975 = 3^2 * 5^2 * 31 and phi(6975) = 60^2.

%p filter:= n -> issqr(numtheory:-phi(n)) and nops

%p (numtheory:-factorset(n))=3:

%p select(filter, [$1..2000]); # after _Robert Israel_ in A324745

%t Select[Range[2000], And[PrimeNu@ # == 3, IntegerQ@ Sqrt@ EulerPhi@ #] &] (* _Michael De Vlieger_, Mar 31 2019 *)

%o (PARI) isok(n) = (omega(n)==3) && issquare(eulerphi(n)); \\ _Michel Marcus_, Mar 19 2019

%Y Intersection of A033992 and A039770.

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

%K nonn

%O 1,1

%A _Bernard Schott_, Mar 16 2019

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Last modified November 14 01:24 EST 2019. Contains 329108 sequences. (Running on oeis4.)