A245664 Prime-partitionable numbers a(n) for which there exists a 2-partition of the set of primes < a(n) that has one subset containing two primes only.

16, 34, 36, 66, 70, 78, 88, 92, 100, 120, 124, 144, 154, 160, 162, 186, 210, 216, 248, 250, 256, 260, 262, 268, 300, 330, 336, 340, 342, 366, 378, 394, 396, 404, 428, 474, 484, 486, 512, 520, 538, 552, 574, 582, 630, 636, 640, 696, 700, 706, 708, 714, 718, 722

Offset: 1

Author

Christopher Hunt Gribble, Jul 28 2014

Keywords

• nonn

Comments

Prime-partitionable numbers are defined in A059756.

To demonstrate that a number is prime-partitionable a suitable 2-partition {P1, P2} of the set of primes < a(n) must be found. In this sequence we are interested in prime-partitionable numbers such that P1 contains 2 odd primes.

Conjecture: If P1 = {p1a, p1b} with p1a and p1b odd primes, p1a < p1b and p1b = 2*k*p1a + 1 for some natural k such that 2*k <= p1a - 3 and if m = p1a + p1b then m is prime-partitionable and belongs to {a(n)}.

Examples

			a(1) = 16 because A059756(1) = 16 and the 2-partition {5, 11}, {2, 3, 7, 13} of the set of primes < 16 demonstrates it.
		

Links

• Christopher Hunt Gribble, Table of n, a(n) for n = 1..145
• Christopher Hunt Gribble, Demonstrating 2-partitions.
• Christopher Hunt Gribble, Conjectured sequence: 20000 terms
• Christopher Hunt Gribble, MAPLE program generating {a(n)}.
• Christopher Hunt Gribble, MAPLE program generating 20000 terms of conjectured sequence.
• W. Holsztynski, R. F. E. Strube, Paths and circuits in finite groups, Discr. Math. 22 (1978) 263-272.
• R. J. Mathar and M. F. Hasler, Is 52 prime-partitionable?, Seqfan thread (Jun 29 2014), arXiv:1510.07997
• W. T. Trotter, Jr. and Paul Erdős, When the Cartesian product of directed cycles is Hamiltonian, J. Graph Theory 2 (1978) 137-142 DOI:10.1002/jgt.3190020206.

Crossrefs

Cf. A059756, A244640.

Programs

Maple

See Gribble links referring to "MAPLE program generating {a(n)}" and "MAPLE program generating 20000 terms of conjectured sequence."

PARI

prime_part(n)=
{
  my (P = primes(primepi(n-1)));
  for (k1 = 2, #P - 1,
    for (k2 = 1, k1 - 1,
      mask = 2^k1 + 2^k2;
      P1 = vecextract(P, mask);
      P2 = setminus(P, P1);
      for (n1 = 1, n - 1,
        bittest(n - n1, 0) || next;
        setintersect(P1, factor(n1)[,1]~) && next;
        setintersect(P2, factor(n-n1)[,1]~) && next;
        next(2)
      );
      print(n, ", ");
    );
  );
}
forstep(m=2,2000,2,prime_part(m));