

A306908


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


4



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
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OFFSET

1,1


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.


LINKS

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


FORMULA

1st family: The primitive terms are p*q*r with p,q,r primes and phi(p*q*r) = (p1)*(q1)*(r1) = 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*(p1)*(q1)*(r1) = 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^(s1) * 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*(p1)*(q1)*(r1) = 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^(s1) * q^(t1) * r^u * m)^2.


EXAMPLE

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.


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 *)


PROG

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


CROSSREFS

Intersection of A033992 and A039770.
Cf. A002496, A054755 (only one prime factor), A324745, A324746, A324747 (two prime factors).
Sequence in context: A217738 A182683 A174601 * A252961 A252962 A296767
Adjacent sequences: A306905 A306906 A306907 * A306909 A306910 A306911


KEYWORD

nonn


AUTHOR

Bernard Schott, Mar 16 2019


STATUS

approved



