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.

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

Author

Christopher Hunt Gribble, Jun 23 2015

Keywords

• nonn

Comments

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.

Examples

			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.
		

Links

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

Crossrefs

Cf. A059756, A245664.

Programs

Maple

# 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):