cp's OEIS Frontend

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.

%I A249302 #40 Dec 23 2024 14:53:44
%S A249302 22,130,222,246,280,286,288,320,324,326,356,416,426,454,470,494,516,
%T A249302 528,556,590,612,634,670,690,738,746,804,818,836,838,870,900,902,904,
%U A249302 922,936,1002,1026,1074,1106,1116,1140,1144,1150,1206,1208,1262,1264,1326,1338
%N 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.
%C A249302 Prime-partitionable numbers are defined in A059756.
%C A249302 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.
%C A249302 Conjecture:
%C A249302 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)}:
%C A249302 m1 = v1*p1a + 1   = v2*p1b + p1a = p1c + p1b
%C A249302 m2 = v3*p1a + 1   = p1b + p1a^2  = p1c + p1a
%C A249302 m3 = v4*p1a + p1b = v5*p1b + 1   = p1c + p1a
%C A249302 where v1, v2, v3, v4 and v5 are odd naturals.
%H A249302 Christopher Hunt Gribble, <a href="/A249302/b249302.txt">Table of n, a(n) for n = 1..77</a>
%H A249302 Christopher Hunt Gribble, <a href="/A249302/a249302.txt">Prime-partitionable numbers with min(#P1)=3</a>
%H A249302 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 A249302 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), <a href="https://arxiv.org/abs/1510.07997">arXiv:1510.07997</a>
%H A249302 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 A249302 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.
%o A249302 (PARI)
%o A249302 prime_part(n)=
%o A249302 {
%o A249302   my (P = primes(primepi(n-1)));
%o A249302   for (k1 = 2, #P - 1,
%o A249302     for (k2 = 1, k1 - 1,
%o A249302       for (k3 = 1, k2 - 1,
%o A249302         mask = 2^k1 + 2^k2 + 2^k3;
%o A249302         P1 = vecextract(P, mask);
%o A249302         P2 = setminus(P, P1);
%o A249302         for (n1 = 1, n - 1,
%o A249302           bittest(n - n1, 0) || next;
%o A249302           setintersect(P1, factor(n1)[,1]~) && next;
%o A249302           setintersect(P2, factor(n-n1)[,1]~) && next;
%o A249302           next(2)
%o A249302         );
%o A249302         print1(n, ", ");
%o A249302       );
%o A249302     );
%o A249302   );
%o A249302 }
%o A249302 # PP = {{2x, x = 1:1000} - {A245664(n), 1:145}}
%o A249302 PP=[2, 4, 6, 8, 10, 12, 14, 18, 20, 22, 24, 26, 28, 30, \
%o A249302     32, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, \
%o A249302     ...
%o A249302     1980, 1984, 1986, 1988, 1990, 1994, 1996, 1998, 2000];
%o A249302 for(m=1,#PP,prime_part(PP[m]));
%Y A249302 Cf. A059756, A244640, A245664.
%K A249302 nonn
%O A249302 1,1
%A A249302 _Christopher Hunt Gribble_, Oct 24 2014