

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



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



KEYWORD

nonn


AUTHOR



STATUS

approved



