This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A244640 #42 Dec 23 2024 14:53:43
%S A244640 2,4,4,16,16,16,8,192,240,128,512,36,24,224,96,896
%N 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.
%C A244640 The sequence comprises the number of all possible partitions {P1,P2} for which each n is prime-partitionable.
%H A244640 Christopher Hunt Gribble, <a href="/A244640/a244640.txt">List of 2-partitions</a>
%H A244640 W. Holsztynski, R. F. E. Strube, <a href="http://dx.doi.org/10.1016/0012-365X(78)90059-6">Paths and circuits in finite groups</a>, Discr. Math. 22 (1978) 263-272.
%H A244640 R. J. Mathar and M. F. Hasler, <a href="https://web.archive.org/web/*/http://list.seqfan.eu/oldermail/seqfan/2014-June/013267.html">Is 52 prime-partitionable?</a>, Seqfan thread (Jun 29 2014)
%H A244640 W. T. Trotter, Jr. and Paul Erdős, <a href="https://www.renyi.hu/~p_erdos/1978-49.pdf">When the Cartesian product of directed cycles is Hamiltonian</a>, J. Graph Theory 2 (1978) 137-142 DOI:10.1002/jgt.3190020206.
%e A244640 Consider the first prime-partitionable number, A059756(1) = 16.
%e A244640 We have P = {2, 3, 5, 7, 11, 13}.
%e A244640 a(1) = 2 because the 2-partitions of P for which 16 is prime-partitionable are:
%e A244640 P1a = {2, 5, 11}, P2a = {3, 7, 13}
%e A244640 P1b = {2, 3, 7, 13}, P2b = {5, 11}
%e A244640 as is shown below:
%e A244640 n1 n2 p1a p2a p1b p2b
%e A244640 1 + 15 - 3 - 5
%e A244640 2 + 14 2 7 2 -
%e A244640 3 + 13 - 13 3 -
%e A244640 4 + 12 2 3 2 -
%e A244640 5 + 11 5 - - 11
%e A244640 6 + 10 2 - 2 5
%e A244640 7 + 9 - 3 7 -
%e A244640 8 + 8 2 - 2 -
%e A244640 9 + 7 - 7 3 -
%e A244640 10 + 6 2 3 2 -
%e A244640 11 + 5 11 - - 5
%e A244640 12 + 4 2 - 2 -
%e A244640 13 + 3 - 3 13 -
%e A244640 14 + 2 2 - 2 -
%e A244640 15 + 1 5 - 3 -
%p A244640 Derived from the program provided by _Richard J. Mathar_ in the second link.
%p A244640 ppartabl := proc (n)
%p A244640 local i, j, pless, p1, p2, n1, n2, pset1, pset2, alln1n2done, foundp1p2;
%p A244640 # construct set of primes < n in pless.
%p A244640 pless := {};
%p A244640 for i from 2 to n-1 do
%p A244640 if isprime(i) then
%p A244640 pless := `union`(pless, {i});
%p A244640 end if;
%p A244640 end do;
%p A244640 # loop over all nontrivial (nonempty) subsets of the primes, P1.
%p A244640 j := 0;
%p A244640 for pset1 in combinat[choose](pless) do
%p A244640 if 1 <= nops(pset1) then
%p A244640 if pset1 = pset2 then
%p A244640 break;
%p A244640 end if;
%p A244640 # P2 is P \ P1.
%p A244640 pset2 := `minus`(pless, pset1);
%p A244640 # flag to indicate that for each n1,n2 we found a pair.
%p A244640 alln1n2done := true;
%p A244640 for n1 to n-1 do
%p A244640 n2 := n-n1;
%p A244640 # flag that we found a (p1,p2).
%p A244640 foundp1p2 := false;
%p A244640 for p1 in pset1 do
%p A244640 if igcd(n1, p1) <> 1 then
%p A244640 foundp1p2 := true;
%p A244640 break;
%p A244640 end if;
%p A244640 for p2 in pset2 do
%p A244640 if igcd(n2, p2) <> 1 then
%p A244640 foundp1p2 := true;
%p A244640 break;
%p A244640 end if;
%p A244640 end do:
%p A244640 if foundp1p2 = true then
%p A244640 break;
%p A244640 end if;
%p A244640 end do:
%p A244640 if foundp1p2 = false then
%p A244640 alln1n2done := false;
%p A244640 break;
%p A244640 end if;
%p A244640 end do:
%p A244640 if alln1n2done = true then
%p A244640 j := j+1;
%p A244640 if j = 1 then
%p A244640 printf("%d\n", n);
%p A244640 end if;
%p A244640 print(j, pset1, pset2);
%p A244640 end if;
%p A244640 end if;
%p A244640 end do:
%p A244640 end proc:
%p A244640 L := [16, 22, 34, 36, 46, 56, 64, 66, 70, 76, 78, 86, 88, 92,
%p A244640 94, 96];
%p A244640 for i from 1 to 16 do
%p A244640 ppartabl(L[i]);
%p A244640 end do:
%Y A244640 Cf. A059756.
%K A244640 nonn
%O A244640 1,1
%A A244640 _Christopher Hunt Gribble_, Jul 03 2014