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A324745
Numbers k with exactly two distinct prime factors and such that phi(k) is a square.
4
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
1,1
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.
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.
EXAMPLE
1st family: 136 = 2^3 * 37 and phi(136) = 8^2.
2nd family: 652 = 2^2 * 163 and phi(652) = 18^2.
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 *)
PROG
(PARI) isok(n) = (omega(n)==2) && issquare(eulerphi(n)); \\ Michel Marcus, Mar 17 2019
CROSSREFS
Intersection of A007774 and A039770.
Sequence in context: A087217 A044972 A132313 * A367148 A267393 A349160
KEYWORD
nonn
AUTHOR
Bernard Schott, Mar 12 2019
STATUS
approved