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

1,1

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)}.

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.

Example

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.

Maple

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

Prog

(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));

Crossrefs

Cf. A059756, A244640.

Sequence in context: A070590 A132370 A185467 * A091216 A350522 A132760

Adjacent sequences: A245661 A245662 A245663 * A245665 A245666 A245667

Keyword

nonn

Author

Christopher Hunt Gribble, Jul 28 2014

Status

approved