A244640 -id:A244640 - cp's OEIS frontend

A245664 Prime-partitionable numbers a(n) for which there exists a 2-partition of the set of primes < a(n) that has one subset containing two primes only.

16, 34, 36, 66, 70, 78, 88, 92, 100, 120, 124, 144, 154, 160, 162, 186, 210, 216, 248, 250, 256, 260, 262, 268, 300, 330, 336, 340, 342, 366, 378, 394, 396, 404, 428, 474, 484, 486, 512, 520, 538, 552, 574, 582, 630, 636, 640, 696, 700, 706, 708, 714, 718, 722

Offset: 1

Christopher Hunt Gribble, Jul 28 2014

nonn

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 P1 contains 2 odd primes.
Conjecture: If P1 = {p1a, p1b} with p1a and p1b odd primes, p1a < p1b and p1b = 2*k*p1a + 1 for some natural k such that 2*k <= p1a - 3 and if m = p1a + p1b then m is prime-partitionable and belongs to {a(n)}.

			a(1) = 16 because A059756(1) = 16 and the 2-partition {5, 11}, {2, 3, 7, 13} of the set of primes < 16 demonstrates it.

Christopher Hunt Gribble, Table of n, a(n) for n = 1..145
Christopher Hunt Gribble, Demonstrating 2-partitions.
Christopher Hunt Gribble, Conjectured sequence: 20000 terms
Christopher Hunt Gribble, MAPLE program generating {a(n)}.
Christopher Hunt Gribble, MAPLE program generating 20000 terms of conjectured sequence.
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.

Cf. A059756, A244640.

See Gribble links referring to "MAPLE program generating {a(n)}" and "MAPLE program generating 20000 terms of conjectured sequence."

prime_part(n)=
{
  my (P = primes(primepi(n-1)));
  for (k1 = 2, #P - 1,
    for (k2 = 1, k1 - 1,
      mask = 2^k1 + 2^k2;
      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)
      );
      print(n, ", ");
    );
  );
}
forstep(m=2,2000,2,prime_part(m));

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.

Original entry on oeis.org

22, 130, 222, 246, 280, 286, 288, 320, 324, 326, 356, 416, 426, 454, 470, 494, 516, 528, 556, 590, 612, 634, 670, 690, 738, 746, 804, 818, 836, 838, 870, 900, 902, 904, 922, 936, 1002, 1026, 1074, 1106, 1116, 1140, 1144, 1150, 1206, 1208, 1262, 1264, 1326, 1338

Offset: 1

Christopher Hunt Gribble, Oct 24 2014

nonn

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 3 odd primes.
Conjecture:
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)}:
m1 = v1*p1a + 1   = v2*p1b + p1a = p1c + p1b
m2 = v3*p1a + 1   = p1b + p1a^2  = p1c + p1a
m3 = v4*p1a + p1b = v5*p1b + 1   = p1c + p1a
where v1, v2, v3, v4 and v5 are odd naturals.

			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.

Christopher Hunt Gribble, Table of n, a(n) for n = 1..77
Christopher Hunt Gribble, Prime-partitionable numbers with min(#P1)=3
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.

Cf. A059756, A244640, A245664.

prime_part(n)=
{
  my (P = primes(primepi(n-1)));
  for (k1 = 2, #P - 1,
    for (k2 = 1, k1 - 1,
      for (k3 = 1, k2 - 1,
        mask = 2^k1 + 2^k2 + 2^k3;
        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), 1:145}}
PP=[2, 4, 6, 8, 10, 12, 14, 18, 20, 22, 24, 26, 28, 30, \
    32, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, \
    ...
    1980, 1984, 1986, 1988, 1990, 1994, 1996, 1998, 2000];
for(m=1,#PP,prime_part(PP[m]));

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.

Original entry on oeis.org

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

Christopher Hunt Gribble, Nov 12 2014

nonn

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.

			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.

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.

Cf. A059756, A244640, A245664, A249302.

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]));

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A245664 Prime-partitionable numbers a(n) for which there exists a 2-partition of the set of primes < a(n) that has one subset containing two primes only.

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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.

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Crossrefs

Programs

PARI

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI