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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. 1
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 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. LINKS Christopher Hunt Gribble, Table of n, a(n) for n = 1..9592 EXAMPLE 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. 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): CROSSREFS Cf. A059756, A245664. Sequence in context: A082127 A031354 A307486 * A200777 A333537 A227827 Adjacent sequences: A259298 A259299 A259300 * A259302 A259303 A259304 KEYWORD nonn AUTHOR Christopher Hunt Gribble, Jun 23 2015 STATUS approved

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Last modified April 23 19:56 EDT 2024. Contains 371916 sequences. (Running on oeis4.)