A324745 Numbers k with exactly two distinct prime factors and such that phi(k) is a square.

10, 12, 34, 40, 48, 57, 63, 74, 76, 85, 108, 136, 160, 185, 192, 202, 219, 250, 292, 296, 304, 394, 432, 451, 489, 505, 513, 514, 544, 567, 629, 640, 652, 679, 768, 802, 808, 873, 972, 985, 1000, 1057, 1154, 1168, 1184, 1216, 1285, 1354

Offset: 1

Author

Bernard Schott, Mar 12 2019

Keywords

• nonn

Comments

This sequence is the intersection of A007774 and A039770.

The sequences A324746 and A324747 form a partition of this sequence.

See the file "Subfamilies and subsequences" (& II) in A039770 for more details, proofs with data, comments, formulas and examples.

The integers with only one prime factor and whose totient is a square are in A054755.

Examples

			1st family: 136 = 2^3 * 37 and phi(136) = 8^2.
2nd family: 652 = 2^2 * 163 and phi(652) = 18^2.
		

Crossrefs

Intersection of A007774 and A039770.

Cf. A054755, A062732, A221285, A324746, A324747.

Programs

Maple

filter:= n -> issqr(numtheory:-phi(n)) and nops(numtheory:-factorset(n))=2:
select(filter, [$1..2000]); # Robert Israel, Mar 18 2019

Mathematica

Select[Range[1400], And[PrimeNu[#] == 2, IntegerQ@ Sqrt@ EulerPhi@ #] &] (* Michael De Vlieger, Mar 21 2019 *)

PARI

isok(n) = (omega(n)==2) && issquare(eulerphi(n)); \\ Michel Marcus, Mar 17 2019

Formula

1st family (A324746): The primitive terms are defined by p*q, p < q, with phi(p*q) = (p-1)*(q-1) = m^2. The general terms are defined by k = p^(2s+1) * q^(2t+1), s,t >= 0, with phi(k) = (p^s * q^t * m)^2.

2nd family (A324747): The primitive terms are defined by p^2 * q, p <> q, with phi(p^2 * q) = p*(p-1)*(q-1) = m^2. The general terms are defined by k = p^(2s ) * q^(2t+1), s >= 1, t >= 0, with phi(k) = (p^(s-1) * q^t * m)^2.