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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. 5
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 (list; graph; refs; listen; history; text; internal format)
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 A132760 A209377

Adjacent sequences:  A245661 A245662 A245663 * A245665 A245666 A245667

KEYWORD

nonn

AUTHOR

Christopher Hunt Gribble, Jul 28 2014

STATUS

approved

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Last modified December 4 05:56 EST 2021. Contains 349470 sequences. (Running on oeis4.)