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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. 2
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 (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 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.

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.

EXAMPLE

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.

PROG

(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]));

CROSSREFS

Cf. A059756, A244640, A245664.

Sequence in context: A309923 A215626 A125247 * A095694 A233060 A299520

Adjacent sequences:  A249299 A249300 A249301 * A249303 A249304 A249305

KEYWORD

nonn

AUTHOR

Christopher Hunt Gribble, Oct 24 2014

STATUS

approved

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Last modified October 22 22:34 EDT 2019. Contains 328335 sequences. (Running on oeis4.)