A249302 Prime-partitionable numbers a(n) for which there exists a 2-partition of the set of primes < a(n) that has a smallest subset containing three primes only.

22, 130, 222, 246, 280, 286, 288, 320, 324, 326, 356, 416, 426, 454, 470, 494, 516, 528, 556, 590, 612, 634, 670, 690, 738, 746, 804, 818, 836, 838, 870, 900, 902, 904, 922, 936, 1002, 1026, 1074, 1106, 1116, 1140, 1144, 1150, 1206, 1208, 1262, 1264, 1326, 1338

Offset: 1

Author

Christopher Hunt Gribble, Oct 24 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 the smallest P1 contains 3 odd primes.

Conjecture:

If P1 = {p1a, p1b, p1c} with p1a, p1b and p1c odd primes and p1a < p1b < p1c then the union of the integer solutions to the three equation groups below, {{m1}, {m2}, {m3}}, contains all even members of {a(n)}:

m1 = v1*p1a + 1 = v2*p1b + p1a = p1c + p1b

m2 = v3*p1a + 1 = p1b + p1a^2 = p1c + p1a

m3 = v4*p1a + p1b = v5*p1b + 1 = p1c + p1a

where v1, v2, v3, v4 and v5 are odd naturals.

Examples

			a(1) = 22 because A059756(2) = 22 and both the 2-partitions {3, 13, 19}, {2, 3, 11, 13, 19} and {5, 7, 17}, {2, 5, 7, 11, 17} of the set of primes < 22 demonstrate it.
		

Links

• Christopher Hunt Gribble, Table of n, a(n) for n = 1..77
• Christopher Hunt Gribble, Prime-partitionable numbers with min(#P1)=3
• 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, A245664.

Programs

PARI

prime_part(n)=
{
  my (P = primes(primepi(n-1)));
  for (k1 = 2, #P - 1,
    for (k2 = 1, k1 - 1,
      for (k3 = 1, k2 - 1,
        mask = 2^k1 + 2^k2 + 2^k3;
        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)
        );
        print1(n, ", ");
      );
    );
  );
}
# PP = {{2x, x = 1:1000} - {A245664(n), 1:145}}
PP=[2, 4, 6, 8, 10, 12, 14, 18, 20, 22, 24, 26, 28, 30, \
    32, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, \
    ...
    1980, 1984, 1986, 1988, 1990, 1994, 1996, 1998, 2000];
for(m=1,#PP,prime_part(PP[m]));