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

1,1

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).

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

Example

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).

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 *)

Crossrefs

Cf. A001262, A141232, A126614, A143012.

Sequence in context: A249568 A106876 A034477 * A246758 A342632 A084643

Adjacent sequences: A140800 A140801 A140802 * A140804 A140805 A140806

Keyword

nonn

Author

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

Extensions

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

Status

approved