cp's OEIS Frontend

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

%I A306908 #38 Apr 11 2019 09:25:20
%S A306908 60,114,126,170,204,240,273,285,315,364,370,380,438,444,456,468,504,
%T A306908 540,680,816,825,902,960,969,978,1010,1026,1071,1095,1100,1134,1212,
%U A306908 1258,1292,1358,1456,1460,1480,1500,1520,1729,1746,1752,1776,1824,1836,1872
%N A306908 Numbers k with exactly three distinct prime factors and such that phi(k) is a square.
%C A306908 This sequence is the intersection of A033992 and A039770.
%C A306908 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 A306908 Robert Israel, <a href="/A306908/b306908.txt">Table of n, a(n) for n = 1..10000</a>
%H A306908 Bernard Schott, <a href="/A306908/a306908.pdf">Subfamilies and subsequences</a>
%F A306908 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 A306908 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 A306908 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 A306908 1st family: 273 = 3 * 7 * 13 and phi(273) = 12^2.
%e A306908 2nd family: 816 = 2^4 * 3 * 17 and phi(816) = 16^2.
%e A306908 3rd family: 6975 = 3^2 * 5^2 * 31 and phi(6975) = 60^2.
%p A306908 filter:= n -> issqr(numtheory:-phi(n)) and nops
%p A306908 (numtheory:-factorset(n))=3:
%p A306908 select(filter, [$1..2000]); # after _Robert Israel_ in A324745
%t A306908 Select[Range[2000], And[PrimeNu@ # == 3, IntegerQ@ Sqrt@ EulerPhi@ #] &] (* _Michael De Vlieger_, Mar 31 2019 *)
%o A306908 (PARI) isok(n) = (omega(n)==3) && issquare(eulerphi(n)); \\ _Michel Marcus_, Mar 19 2019
%Y A306908 Intersection of A033992 and A039770.
%Y A306908 Cf. A002496, A054755 (only one prime factor), A324745, A324746, A324747 (two prime factors).
%K A306908 nonn
%O A306908 1,1
%A A306908 _Bernard Schott_, Mar 16 2019