A244640 a(n) is the number of 2-partitions of the set of primes less than A059756(n) that demonstrate that A059756(n) is prime-partitionable.

2, 4, 4, 16, 16, 16, 8, 192, 240, 128, 512, 36, 24, 224, 96, 896

Offset: 1

Author

Christopher Hunt Gribble, Jul 03 2014

Keywords

• nonn

Comments

The sequence comprises the number of all possible partitions {P1,P2} for which each n is prime-partitionable.

Examples

			Consider the first prime-partitionable number, A059756(1) = 16.
We have P = {2, 3, 5, 7, 11, 13}.
a(1) = 2 because the 2-partitions of P for which 16 is prime-partitionable are:
    P1a = {2, 5, 11},       P2a = {3, 7, 13}
    P1b = {2, 3, 7, 13},    P2b = {5, 11}
as is shown below:
       n1   n2   p1a   p2a      p1b   p2b
        1 + 15     -     3        -     5
        2 + 14     2     7        2     -
        3 + 13     -    13        3     -
        4 + 12     2     3        2     -
        5 + 11     5     -        -    11
        6 + 10     2     -        2     5
        7 +  9     -     3        7     -
        8 +  8     2     -        2     -
        9 +  7     -     7        3     -
       10 +  6     2     3        2     -
       11 +  5    11     -        -     5
       12 +  4     2     -        2     -
       13 +  3     -     3       13     -
       14 +  2     2     -        2     -
       15 +  1     5     -        3     -
		

Links

• Christopher Hunt Gribble, List of 2-partitions
• 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)
• 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.

Programs

Maple

Derived from the program provided by Richard J. Mathar in the second link.
ppartabl := proc (n)
  local i, j, pless, p1, p2, n1, n2, pset1, pset2, alln1n2done, foundp1p2;
  # construct set of primes < n in pless.
  pless := {};
  for i from 2 to n-1 do
    if isprime(i) then
      pless := `union`(pless, {i});
    end if;
  end do;
  # loop over all nontrivial (nonempty) subsets of the primes, P1.
  j := 0;
  for pset1 in combinat[choose](pless) do
    if 1 <= nops(pset1) then
      if pset1 = pset2 then
        break;
      end if;
      # P2 is P \ P1.
      pset2 := `minus`(pless, pset1);
      # flag to indicate that for each n1,n2 we found a pair.
      alln1n2done := true;
      for n1 to n-1 do
        n2 := n-n1;
        # flag that we found a (p1,p2).
        foundp1p2 := false;
        for p1 in pset1 do
          if igcd(n1, p1) <> 1 then
            foundp1p2 := true;
            break;
          end if;
          for p2 in pset2 do
            if igcd(n2, p2) <> 1 then
              foundp1p2 := true;
              break;
            end if;
          end do:
          if foundp1p2 = true then
            break;
          end if;
        end do:
        if foundp1p2 = false then
          alln1n2done := false;
          break;
        end if;
      end do:
      if alln1n2done = true then
        j := j+1;
        if j = 1 then
          printf("%d\n", n);
        end if;
        print(j, pset1, pset2);
      end if;
    end if;
  end do:
end proc:
L := [16, 22, 34, 36, 46, 56, 64, 66, 70, 76, 78, 86, 88, 92,
      94, 96];
for i from 1 to 16 do
  ppartabl(L[i]);
end do: