

A245664


Primepartitionable numbers a(n) for which there exists a 2partition 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
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OFFSET

1,1


COMMENTS

Primepartitionable numbers are defined in A059756.
To demonstrate that a number is primepartitionable a suitable 2partition {P1, P2} of the set of primes < a(n) must be found. In this sequence we are interested in primepartitionable 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 primepartitionable and belongs to {a(n)}.


LINKS

Christopher Hunt Gribble, Table of n, a(n) for n = 1..145
Christopher Hunt Gribble, Demonstrating 2partitions.
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) 263272.
R. J. Mathar and M. F. Hasler, Is 52 primepartitionable?, 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) 137142 DOI:10.1002/jgt.3190020206.


EXAMPLE

a(1) = 16 because A059756(1) = 16 and the 2partition {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(n1)));
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(nn1)[, 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



