cp's OEIS Frontend

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

%I A324745 #33 Mar 22 2019 10:07:13
%S A324745 10,12,34,40,48,57,63,74,76,85,108,136,160,185,192,202,219,250,292,
%T A324745 296,304,394,432,451,489,505,513,514,544,567,629,640,652,679,768,802,
%U A324745 808,873,972,985,1000,1057,1154,1168,1184,1216,1285,1354
%N A324745 Numbers k with exactly two distinct prime factors and such that phi(k) is a square.
%C A324745 This sequence is the intersection of A007774 and A039770.
%C A324745 The sequences A324746 and A324747 form a partition of this sequence.
%C A324745 See the file "Subfamilies and subsequences" (& II) in A039770 for more details, proofs with data, comments, formulas and examples.
%C A324745 The integers with only one prime factor and whose totient is a square are in A054755.
%F A324745 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.
%F A324745 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.
%e A324745 1st family: 136 = 2^3 * 37 and phi(136) = 8^2.
%e A324745 2nd family: 652 = 2^2 * 163 and phi(652) = 18^2.
%p A324745 filter:= n -> issqr(numtheory:-phi(n)) and nops(numtheory:-factorset(n))=2:
%p A324745 select(filter, [$1..2000]); # _Robert Israel_, Mar 18 2019
%t A324745 Select[Range[1400], And[PrimeNu[#] == 2, IntegerQ@ Sqrt@ EulerPhi@ #] &] (* _Michael De Vlieger_, Mar 21 2019 *)
%o A324745 (PARI) isok(n) = (omega(n)==2) && issquare(eulerphi(n)); \\ _Michel Marcus_, Mar 17 2019
%Y A324745 Intersection of A007774 and A039770.
%Y A324745 Cf. A054755, A062732, A221285, A324746, A324747.
%K A324745 nonn
%O A324745 1,1
%A A324745 _Bernard Schott_, Mar 12 2019