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
KEYWORD
nonn
AUTHOR
Christopher Hunt Gribble, Jul 28 2014
STATUS
approved