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 A102605 Number of ways of writing 2n+1 as p+q+r where p,q,r are primes with p < q < r, offset=0. 4
 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 2, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 6, 6, 9, 8, 8, 11, 11, 10, 13, 13, 12, 14, 15, 13, 18, 17, 14, 21, 19, 17, 25, 20, 21, 26, 25, 22, 30, 28, 21, 32, 31, 23, 37, 32, 27, 39, 36, 32, 43, 41, 36, 45, 44, 35, 51, 48, 34, 54, 48, 36, 59, 50, 43, 60, 55, 46, 61 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,11 COMMENTS The graph of this function shows two main branches, each with further subdivisions. It seems that the main branches result from the fact that values a(3k+1) are in the mean roughly 30% lower than values a(3k) and a(3k+2). This can be explained by the fact that the sum of 3 primes (with equal probability of being congruent to 1 or to 5 mod 6) is congruent to 3 (mod 6) in only 2 out of 8 cases, and congruent to 1 or to 5 (mod 6) in 3 out of 8 cases, for each of these two residues. Analyzing the frequencies of the possible residues mod 30 explains the further sub-branches: A sum of 3 primes is congruent to 1, 3, ..., 29 (mod 30) in (42, 29, 33, 39, 29, 36, 36, 30, 39, 30, 39, 30, 36, 37, 27) out of 512 cases. - M. F. Hasler, Oct 27 2017 LINKS Alois P. Heinz, Table of n, a(n) for n = 0..10000 EXAMPLE a(19) = 6 because 2*19+1 = 39 and 39 = 3+5+31 = 3+7+29 = 3+13+23 = 3+17+19 = 5+11+23 = 7+13+19. PROG (PARI) A102605(n, s=0)={forprime(p=1, (n*=2)\3, my(d=n-p); forprime(q=p+1, d\2, isprime(d+1-q)&&s++)); s} \\ M. F. Hasler, Oct 27 2017 CROSSREFS Number of ways of writing 2n+1 as p+q+r where p, q, r are primes with p <= q <= r gives A054860. Bisection of A125688 (odd part). - Alois P. Heinz, Nov 14 2012 Sequence in context: A020911 A029125 A074990 * A112995 A078452 A263997 Adjacent sequences:  A102602 A102603 A102604 * A102606 A102607 A102608 KEYWORD nonn,look AUTHOR Zak Seidov, Jan 29 2005 STATUS approved

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Last modified June 25 05:41 EDT 2019. Contains 324346 sequences. (Running on oeis4.)