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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.
1

%I #26 Dec 23 2024 14:53:43

%S 46,76,96,106,134,142,146,204,218,276,310,408,438,466,518,534,536,546,

%T 580,624,650,672,680,694,792,800,896,970,1000,1016,1100,1160,1170,

%U 1318,1344,1358,1364,1384,1470,1480

%N 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.

%C Prime-partitionable numbers are defined in A059756.

%C 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.

%C Conjecture:

%C 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)}:

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

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

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

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

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

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

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

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

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

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

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

%H Christopher Hunt Gribble, <a href="/A245372/a245372.txt">Prime-partitionable numbers with #P1 = 4</a>

%H 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 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 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 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.

%o (PARI)

%o prime_part(n)=

%o {

%o my (P = primes(primepi(n-1)));

%o for (k1 = 2, #P - 1,

%o for (k2 = 1, k1 - 1,

%o for (k3 = 1, k2 - 1,

%o for (k4 = 1, k3 - 1,

%o mask = 2^k1 + 2^k2 + 2^k3 + 2^k4;

%o P1 = vecextract(P, mask);

%o P2 = setminus(P, P1);

%o for (n1 = 1, n - 1,

%o bittest(n - n1, 0) || next;

%o setintersect(P1, factor(n1)[,1]~) && next;

%o setintersect(P2, factor(n-n1)[,1]~) && next;

%o next(2)

%o );

%o print1(n, ", ");

%o );

%o );

%o );

%o );

%o }

%o # PP = {{2x, x = 1:1000} - {A245664(n), n = 1:145}

%o # - {A249302(n), n = 1:77}}

%o PP = [2, 4, 6, 8, 10, 12, 14, 18, 20, 24, 26, 28, 30, 32, \

%o 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, \

%o ...

%o 1994, 1996, 1998, 2000];

%o for(m=1,#PP,prime_part(PP[m]));

%Y Cf. A059756, A244640, A245664, A249302.

%K nonn

%O 1,1

%A _Christopher Hunt Gribble_, Nov 12 2014