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A324746 Numbers k with exactly two distinct prime factors and such that phi(k) is square, when k = p^(2s+1) * q^(2t+1) with p < q primes, s,t >= 0.

%I A324746 #32 Apr 03 2019 03:50:39
%S A324746 10,34,40,57,74,85,136,160,185,202,219,250,296,394,451,489,505,513,
%T A324746 514,544,629,640,679,802,808,985,1000,1057,1154,1184,1285,1354,1387,
%U A324746 1417,1576,1717,1971,2005,2047,2056,2125,2176,2509,2560,2594,2649,2761,2885,3097
%N A324746 Numbers k with exactly two distinct prime factors and such that phi(k) is square, when k = p^(2s+1) * q^(2t+1) with p < q primes, s,t >= 0.
%C A324746 An integer belongs to this sequence iff (p-1)*(q-1) = m^2.
%C A324746 This is the first subsequence of A324745, the second one is A324747.
%C A324746 Some values of (k,p,q,m): (10,2,5,2), (34,2,17,4), (40,2,5,4), (57,3,19,4), (74,2,37,6), (85,5,17,8).
%C A324746 The primitive terms of this sequence are the products p * q, with p < q which satisfy (p-1)*(q-1) = m^2; the first few are 10, 34, 57, 74, 85, 185. These primitives form exactly the sequence A247129. Then the integers (p*q) * p^2 and (p*q) * q^2 are new terms of the general sequence.
%C A324746 The number of semiprimes p*q whose totient is a square equal to (2*n)^2 can be found in A306722.
%F A324746 phi(p*q) = (p-1)*(q-1) = m^2 for primitive terms.
%F A324746 phi(k) = (p^s * q^t * m)^2 with k as in the name of this sequence.
%e A324746 629 = 17 * 37 and phi(629) = 16 * 36 = 9^2.
%e A324746 808 = 2^3 * 101 and phi(808) = (2^1 * 101^0 * 10)^2 = 20^2.
%p A324746 N:= 10^4:
%p A324746 Res:= {}:
%p A324746 p:= 1:
%p A324746 do
%p A324746   p:= nextprime(p);
%p A324746   if p^2 >= N then break fi;
%p A324746   F:= ifactors(p-1)[2];
%p A324746   dm:= mul(t[1]^ceil(t[2]/2),t=F);
%p A324746   for j from (p-1)/dm+1 do
%p A324746     q:= (j*dm)^2/(p-1) + 1;
%p A324746     if q > N then break fi;
%p A324746     if isprime(q) then Res:= Res union {seq(seq(
%p A324746       p^(2*s+1)*q^(2*t+1),t=0..floor((log[q](N/p^(2*s+1))-1)/2)),
%p A324746       s=0..floor((log[p](N/q)-1)/2))} fi
%p A324746   od
%p A324746 od:
%p A324746 sort(convert(Res,list)); # _Robert Israel_, Mar 22 2019
%t A324746 Select[Range[6, 3100], And[PrimeNu@ # == 2, IntegerQ@ Sqrt@ EulerPhi@ #, IntegerQ@ Sqrt[Times @@ (FactorInteger[#][[All, 1]] - 1 )]] &] (* _Michael De Vlieger_, Mar 24 2019 *)
%o A324746 (PARI) isok(k) = {if (issquare(eulerphi(k)), my(expo = factor(k)[,2]); if ((#expo == 2)&& (expo[1]%2) == (expo[2]%2), return (1)););} \\ _Michel Marcus_, Mar 18 2019
%Y A324746 Cf. A039770, A062732, A221285, A054755, A324745, A324747, A306908.
%Y A324746 Cf. A306722, A247129 (subsequence of primitives).
%K A324746 nonn
%O A324746 1,1
%A A324746 _Bernard Schott_, Mar 12 2019