A245664 -id:A245664 - 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.

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]));

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

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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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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.

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

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