A245372 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 four primes only.

46, 76, 96, 106, 134, 142, 146, 204, 218, 276, 310, 408, 438, 466, 518, 534, 536, 546, 580, 624, 650, 672, 680, 694, 792, 800, 896, 970, 1000, 1016, 1100, 1160, 1170, 1318, 1344, 1358, 1364, 1384, 1470, 1480

Offset: 1

Author

Christopher Hunt Gribble, Nov 12 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 4 odd primes.

Conjecture:

If P1 = {p1a, p1b, p1c, p1d} with p1a, p1b, p1c and p1d odd primes and p1a < p1b < p1c < p1d then the union of the integer solutions to the ten equation groups below, {{m1}, {m2}, {m3}, {m4}, {m5}, {m6}, {m7}, {m8}, {m9}, {m10}}, contains all even members of {a(n)}:

m1 = v1*p1a+1 = v2*p1b+p1a = v3*p1c+p1b = v4*p1d+p1c

m2 = v5*p1a+1 = v6*p1b+p1a^2 = v7*p1c+p1b = v8*p1d+p1a

m3 = v9*p1a+1 = v10*p1b+p1a^3 = v11*p1c+p1a^2 = v12*p1d+p1a

m4 = v13*p1a+1 = v14*p1b+p1c = v15*p1c+p1a = v16*p1d+p1b

m5 = v17*p1a+1 = v18*p1b+p1c = v19*p1c+p1a^2 = v20*p1d+p1a

m6 = v21*p1a+p1b = v22*p1b+1 = v23*p1c+p1a = v24*p1d+p1c

m7 = v25*p1a+p1b = v26*p1b+1 = v27*p1c+p1a^2 = v28*p1d+p1a

m8 = v29*p1a+p1b = v30*p1b+p1c = v31*p1c+1 = v32*p1d+p1a

m9 = v33*p1a+p1c = v34*p1b+1 = v35*p1c+p1b = v36*p1d+p1a

m10 = v37*p1a+p1c = v38*p1b+p1a = v39*p1c+1 = v40*p1d+p1b

where the vi, i = 1..40 are constrained odd naturals.

Examples

			a(1) = 46 because A245602(5) = 46 and the 2-partition {3, 19, 37, 43} {2, 5, 7, 11, 13, 17, 23, 29, 31, 41} of the set of primes < 46 demonstrates it.
		

Links

• Christopher Hunt Gribble, Prime-partitionable numbers with #P1 = 4
• 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, A249302.

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,
        for (k4 = 1, k3 - 1,
          mask = 2^k1 + 2^k2 + 2^k3 + 2^k4;
          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), n = 1:145}
#                        - {A249302(n), n = 1:77}}
PP = [2, 4, 6, 8, 10, 12, 14, 18, 20, 24, 26, 28, 30, 32, \
      38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, \
      ...
      1994, 1996, 1998, 2000];
for(m=1,#PP,prime_part(PP[m]));