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A140803 Numbers of the form (2^(p*q)-1) /((2^p-1)*(2^q-1)), where p>q are primes.

%I A140803 #20 Sep 17 2018 09:36:25
%S A140803 3,11,43,151,683,2359,2731,43691,174763,599479,2796203,8727391,
%T A140803 9588151,178956971,715827883,2454285751,39268347319,45812984491,
%U A140803 567767102431,733007751851,2932031007403,10052678938039,46912496118443,145295143558111,3002399751580331,41175768098368951,192153584101141163
%N A140803 Numbers of the form (2^(p*q)-1) /((2^p-1)*(2^q-1)), where p>q are primes.
%C A140803 The sequence contains, in particular, A126614 (q=2) and A143012 (q=3).
%C A140803 If pq-1 is squarefree then the terms of the sequence are either primes or overpseudoprimes to base 2 (see A141232). In particular, they are strong pseudoprimes to base 2 (A001262).
%H A140803 Robert Israel, <a href="/A140803/b140803.txt">Table of n, a(n) for n = 1..825</a>
%H A140803 V. Shevelev, <a href="https://arxiv.org/abs/0807.2332">Process of "primoverization" of numbers of the form a^n-1</a>, arXiv:0807.2332 [math.NT], 2008.
%H A140803 S. Wagstaff, <a href="http://homes.cerias.purdue.edu/~ssw/cun/third/pmain901">Factorizations of 2^n-1</a>
%e A140803 Entry 3 from (q=2,p=3), entry 11 from (q=2,p=5), entry 43 from (q=2,p=7), entry 151 from (q=3,p=5), entry 683 from (q=2,p=11).
%p A140803 N:= 100: # to use all (p,q) with p*q < N
%p A140803 Primes:= select(isprime,[$2..floor(N/2)]):
%p A140803 A:= {}:
%p A140803 for i from 1 to nops(Primes) do
%p A140803   p:= Primes[i];
%p A140803   Qs:= select(q -> q < N/p, [seq(Primes[j],j=1..i-1)]);
%p A140803   A:= A union {seq((2^(p*q)-1)/(2^p-1)/(2^q-1),q=Qs)};
%p A140803 od:
%p A140803 A; # _Robert Israel_, Sep 02 2014
%t A140803 terms = 27; Clear[seq]; seq[m_] := seq[m] = Table[(2^(p q)-1)/((2^p-1) (2^q-1)), {q, Prime[Range[m]]}, {p, Prime[Range[PrimePi[q]+1, terms]]}] // Flatten // Union // PadRight[#, terms]&;
%t A140803 seq[1]; seq[m=2]; While[seq[m] != seq[m-1], m++]; seq[m] (* _Jean-François Alcover_, Sep 17 2018 *)
%Y A140803 Cf. A001262, A141232, A126614, A143012.
%K A140803 nonn
%O A140803 1,1
%A A140803 _Vladimir Shevelev_, Jul 15 2008, Jul 22 2008; corrected Sep 07 2008
%E A140803 a(17) to a(27) from _Robert Israel_, Sep 03 2014