A059756 -id:A059756 - cp's OEIS frontend

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

Christopher Hunt Gribble, Jul 03 2014

nonn

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

			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     -

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.

Cf. A059756.

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:

A059757 Initial terms of smallest Erdős-Woods intervals corresponding to the terms of A059756.

Original entry on oeis.org

2184, 3521210, 47563752566, 12913165320, 21653939146794, 172481165966593120, 808852298577787631376, 91307018384081053554, 1172783000213391981960, 26214699169906862478864, 27070317575988954996883440, 92274830076590427944007586984, 3061406404565905778785058155412

Offset: 1

Nik Lygeros (webmaster(AT)lygeros.org), Feb 12 2001

nonn

The term a(1)=2184 is the start of the lowest interval with an Erdős-Woods interval length of 16 = A059756(1), and the list of others of that length 16 appear to be the same as A194585. - R. J. Mathar, Jul 02 2014

			For the Erdős-Woods number 16, no number between 2184 and 2184+16 is coprime to both 2184 and 2184+16. 2184 is the smallest such nonnegative integer.

Bertram Felgenhauer, Table of n, a(n) for n = 1..106
Code Golf Stack Exchange user xash, Find the Erdős-Woods origin.
Bertram Felgenhauer, Some OEIS computations (first 106 terms and supporting code)

Cf. A059756, A194585, A244620.

a(5)-a(13) corrected by Peter Kagey, Jul 01 2021, based on the Code Golf Stack Exchange link.

A111042 Odd terms of A059756.

Original entry on oeis.org

903, 2545, 4533, 5067, 8759, 9071, 9269, 10353, 11035, 11625, 11865, 13629, 15395, 15493, 16803, 17955, 18575, 18637, 19149, 24189, 35547, 36941, 37911, 42111, 43613, 45179, 50717, 52383, 53367, 54159, 58285, 59903, 61333, 62373, 65109, 67807, 68483, 70109, 72575

Offset: 1

Victor S. Miller, Oct 08 2005

Dowe (1989) conjectured that all Erdős-Woods numbers (A059756) are even. The first counterexamples were found in 2001 by Marcin Bienkowski, Mirek Korzeniowski and Krysztof Lorys, and independently by Nik Lygeros (Cégielski et al., 2003). - Amiram Eldar, Jun 20 2021

Bertram Felgenhauer, Table of n, a(n) for n = 1..227
Patrick Cégielski, François Heroult and Denis Richard, On the amplitude of intervals of natural numbers whose every element has a common prime divisor with at least an extremity, Theor. Comp. Sci., Vol. 303, No 1 (2003), pp. 53-62.
David L. Dowe, On the existence of sequences of co-prime pairs of integers, J. Austral. Math. Soc. Ser. A, Vol. 47, No. 1 (1989), pp. 84-89.
Bertram Felgenhauer, Some OEIS computations. (Includes the terms of this sequence up to 10^6)
Nik Lygeros, 737 - From the computation of Erdös-Woods Numbers to the quadratic Goldbach Conjecture, 2012.

Cf. A059756.

Corrected by T. D. Noe, Nov 02 2006

More terms from Felgenhauer added by Amiram Eldar, Jun 20 2021

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.

Original entry on oeis.org

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));

A244620 Initial terms of Erdős-Wood intervals of length 22.

Original entry on oeis.org

3521210, 6178458, 13220900, 15878148, 22920590, 25577838, 32620280, 35277528, 42319970, 44977218, 52019660, 54676908, 61719350, 64376598, 71419040, 74076288, 81118730, 83775978, 90818420, 93475668, 100518110, 103175358, 110217800, 112875048, 119917490

Offset: 1

R. J. Mathar, Jul 02 2014

nonn

By definition of the intervals in A059756, these are numbers that start a sequence of 23 consecutive integers such that none of the 23 integers is coprime to the first and also coprime to the last integer of the interval.

Hence each initial term of an Erdős-Wood interval is the initial term of a stapled interval of length A059756(n) + 1 (see definition in A090318). - Christopher Hunt Gribble, Dec 02 2014

			3521210 = 2*5*7*11*17*269 and 3521210+22 = 3521232 = 2^4 * 3^4 * 11 * 13 * 19, and all numbers in [3521210,3521232] have at least one prime factor in {2, 3, 5, 7, 11, 13, 17, 19, 269}. Therefore 3521210 is in the list.

Christopher Hunt Gribble, Table of n, a(n) for n = 1..1000
Wikipedia, Erdős-Woods number
Index entries for linear recurrences with constant coefficients, signature (1, 1, -1).

Cf. A059757, A194585, A090318, A130173.

isEWood := proc(n,ewlength)
    local nend,fsn,fsne,fsall,fsk ;
    nend := n+ewlength ;
    fsn := numtheory[factorset](n) ;
    fsne := numtheory[factorset](nend) ;
    fsall := fsn union fsne ;
    for k from n to nend do
        fsk := numtheory[factorset](k) ;
        if fsk intersect fsall = {} then
            return false;
        end if;
    end do:
    return true;
end proc:
for n from 2 do
    if isEWood(n,22) then
        print(n) ;
    end if;
end do:

a(1) = A059757(2).
From Christopher Hunt Gribble, Dec 02 2014: (Start)
a(1) = A130173(524).
a(2*n+1) = 3521210 + 9699690*n.
a(2*n+2) = 6178458 + 9699690*n.
a(n) = (-4849867 - 2192597*(-1)^n + 9699690*n)/2.
a(n) = a(n-1) + a(n-2) - a(n-3).
G.f.: (3521232*x^2+2657248*x+3521210) / ((x-1)^2*(x+1)). (End)

More terms from Christopher Hunt Gribble, Dec 03 2014

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]));

A342310 Prime Erdős-Woods numbers.

Original entry on oeis.org

15493, 18637, 43613, 45179, 61333, 67807, 68483, 80671, 87383, 120557, 130349, 138077, 139187, 199373, 341027, 485201, 581869, 598079, 776497, 786931, 790063, 869777, 936709, 943013, 964303, 1085737, 1166153, 1187999, 1192207, 1225517, 1363837, 1392959, 1404419

Offset: 1

Jianing Song, Mar 08 2021

A number k is an Erdős-Woods number (A059756) if there exists m such that for every 0 <= i <= m, at least one of gcd(m+i, m) > 1 or gcd(m+i, m+k) > 1 holds. This sequence gives the prime terms in A059756.

Bertram Felgenhauer, Some OEIS computations (includes the terms of this sequence up to 100000).
Jinyuan Wang, C++ program.
Wikipedia, Erdős-Woods number.

Subsequence of A059756 and A111042.

More terms from Jinyuan Wang, Jan 28 2025

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]));

A259301 Taken over all those prime-partitionable numbers m for which there exists a 2-partition of the set of primes < m that has one subset containing two primes only, a(n) is the frequency with which the smaller prime occurs, where n is the prime index.

Original entry on oeis.org

0, 0, 1, 1, 3, 3, 3, 2, 4, 4, 3, 4, 5, 7, 8, 5, 8, 7, 8, 9, 10, 10, 11, 12, 12, 14, 13, 13, 12, 15, 14, 14, 17, 14, 19, 17, 12, 18, 13, 19, 20, 22, 20, 23, 21, 15, 21, 21, 23, 25, 26, 23, 26, 26, 19, 23, 27, 24, 29, 27, 26, 28, 31, 29, 30, 25, 30, 29, 34, 30

Offset: 1

Christopher Hunt Gribble, Jun 23 2015

nonn

A number n is called a prime partitionable number if there is a partition {P1,P2} of the primes less than n such that for any composition n1+n2=n, either there is a prime p in P1 such that p | n1 or there is a prime p in P2 such that p | n2.
To demonstrate that a positive integer m is prime-partitionable, a suitable 2-partition {P1, P2} of the set of primes < m 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 positive integer k such that 2*k <= p1a - 3 and if m = p1a + p1b then m is prime-partitionable.

			The table below shows all p1a and p1b pairs for p1a <= 29 that demonstrate that m is prime-partitionable.
. n    p1a    p1b     2k      m
. 3      5     11      2     16
. 4      7     29      4     36
. 5     11     23      2     34
.       11     67      6     78
.       11     89      8    100
. 6     13     53      4     66
.       13     79      6     92
.       13    131     10    144
. 7     17    103      6    120
.       17    137      8    154
.       17    239     14    256
. 8     19    191     10    210
.       19    229     12    248
. 9     23     47      2     70
.       23    139      6    162
.       23    277     12    300
.       23    461     20    484
.10     29     59      2     88
.       29    233      8    262
.       29    349     12    378
.       29    523     18    552
By examining the p1a column it can be seen that
a(1) = 0, a(2) = 0, a(3) = 1, a(4) = 1, a(5) = 3, a(6) = 3,
a(7) = 3, a(8) = 2, a(9) = 4, a(10) = 4.

Christopher Hunt Gribble, Table of n, a(n) for n = 1..9592

Cf. A059756, A245664.

# Makes use of conjecture in COMMENTS section.
ppgen := proc (ub)
  local freq_p1a, i, j, k, nprimes, p1a, p1b, pless;
  # Construct set of primes < ub in pless.
  pless := {};
  for i from 3 to ub do
    if isprime(i) then
      pless := `union`(pless, {i});
    end if
  end do;
  nprimes := numelems(pless);
  # Determine frequency of each p1a.
  printf("0, ");    # For prime 2.
  for j to nprimes do
    p1a := pless[j];
    freq_p1a := 0;
    for k to (p1a-3)/2 do
      p1b := 2*k*p1a+1;
      if isprime(p1b) then
        freq_p1a := freq_p1a+1;
      end if;
    end do;
    printf("%d, ", freq_p1a);
  end do;
end proc:
ub := 1000:
ppgen(ub):

A259560 Primes p such that p = 2kq + 1 for k a positive integer, q an odd prime and 2k <= q - 3.

Original entry on oeis.org

11, 23, 29, 47, 53, 59, 67, 79, 83, 89, 103, 107, 131, 137, 139, 149, 167, 173, 179, 191, 223, 227, 229, 233, 239, 263, 269, 277, 283, 293, 311, 317, 347, 349, 359, 367, 373, 383, 389, 431, 439, 461, 467, 479, 499, 503, 509, 523, 557, 563, 569, 587, 593, 607

Offset: 1

Christopher Hunt Gribble, Jun 30 2015

nonn

This sequence is associated with the conjecture in A245664 that p + q is prime-partitionable.

There are 138438 values of p in the first 216816 primes, i.e., 63.85%, all of which are distinct.

			The table lists values of n, q, 2k and p for 1 <= n <= 20.
.n      q     2k      p (a(n))
.1      5      2     11
.2     11      2     23
.3      7      4     29
.4     23      2     47
.5     13      4     53
.6     29      2     59
.7     11      6     67
.8     13      6     79
.9     41      2     83
10     11      8     89
11     17      6    103
12     53      2    107
13     13     10    131
14     17      8    137
15     23      6    139
16     37      4    149
17     83      2    167
18     43      4    173
19     89      2    179
20     19     10    191

Christopher Hunt Gribble, Table of n, a(n) for n = 1..20000

Cf. A059756, A245664.

ppgen := proc (n)
  local i, j, k, nprimes, p1a, p1b, p1b_ind, pless, pless_idx, p1b_ind_num_0, p1b_ind_num_1;
  pless := {};
  for i from 3 to n do
    if isprime(i) then
      pless := `union`(pless, {i})
    end if
  end do;
  nprimes := numelems(pless);
  p1b_ind := Vector(nprimes);
  for j to nprimes do
    p1a := pless[j];
    if (1/2)*pless[-1]+1/2 < p1a then
      break
    end if;
    for k to (1/2)*p1a-3/2 do
      p1b := 2*k*p1a+1;
      if member(p1b, pless, 'pless_idx') then
        p1b_ind[pless_idx] := 1
      elif pless[-1] < p1b then
        break
      end if
    end do
  end do;
  p1b_ind_num_0 := 1;
  p1b_ind_num_1 := 0;
  for i to nprimes do
    if p1b_ind[i] = 0 then
      p1b_ind_num_0 := p1b_ind_num_0+1
    else
      p1b_ind_num_1 := p1b_ind_num_1+1;
      fprintf(fop, "%d %d\n", p1b_ind_num_1, pless[i])
    end if
  end do
end proc;
n := 376200;
ppgen(n);

is(n)=my(f=factor(n\2)[,1]); for(i=1,#f, if(n\2/f[i]*2<=f[i]-3, return(isprime(n)))); 0 \\ Charles R Greathouse IV, Jul 15 2015

cp's OEIS Frontend

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.

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A059757 Initial terms of smallest Erdős-Woods intervals corresponding to the terms of A059756.

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A111042 Odd terms of A059756.

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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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A244620 Initial terms of Erdős-Wood intervals of length 22.

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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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A342310 Prime Erdős-Woods numbers.

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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.

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A259301 Taken over all those prime-partitionable numbers m for which there exists a 2-partition of the set of primes < m that has one subset containing two primes only, a(n) is the frequency with which the smaller prime occurs, where n is the prime index.