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

3, 11, 43, 151, 683, 2359, 2731, 43691, 174763, 599479, 2796203, 8727391, 9588151, 178956971, 715827883, 2454285751, 39268347319, 45812984491, 567767102431, 733007751851, 2932031007403, 10052678938039, 46912496118443, 145295143558111, 3002399751580331, 41175768098368951, 192153584101141163

Offset: 1

Author

Vladimir Shevelev, Jul 15 2008, Jul 22 2008; corrected Sep 07 2008

Keywords

• nonn

Comments

The sequence contains, in particular, A126614 (q=2) and A143012 (q=3).

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).

Examples

			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).
		

Links

• Robert Israel, Table of n, a(n) for n = 1..825
• V. Shevelev, Process of "primoverization" of numbers of the form a^n-1, arXiv:0807.2332 [math.NT], 2008.
• S. Wagstaff, Factorizations of 2^n-1

Crossrefs

Cf. A001262, A141232, A126614, A143012.

Programs

Maple

N:= 100: # to use all (p,q) with p*q < N
Primes:= select(isprime,[$2..floor(N/2)]):
A:= {}:
for i from 1 to nops(Primes) do
  p:= Primes[i];
  Qs:= select(q -> q < N/p, [seq(Primes[j],j=1..i-1)]);
  A:= A union {seq((2^(p*q)-1)/(2^p-1)/(2^q-1),q=Qs)};
od:
A; # Robert Israel, Sep 02 2014

Mathematica

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]&;
seq[1]; seq[m=2]; While[seq[m] != seq[m-1], m++]; seq[m] (* Jean-François Alcover, Sep 17 2018 *)

Extensions

a(17) to a(27) from Robert Israel, Sep 03 2014