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

1,1

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 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.

Links

Table of n, a(n) for n=1..40.

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.

Example

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.

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,
        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]));

Crossrefs

Cf. A059756, A244640, A245664, A249302.

Sequence in context: A051413 A167283 A085434 * A020351 A113691 A188598

Adjacent sequences: A245369 A245370 A245371 * A245373 A245374 A245375

Keyword

nonn

Author

Christopher Hunt Gribble, Nov 12 2014

Status

approved