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A102605
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Number of ways of writing 2n+1 as p+q+r where p,q,r are primes with p < q < r, offset=0.
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4
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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
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OFFSET
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0,11
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COMMENTS
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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
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LINKS
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EXAMPLE
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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.
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PROG
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(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
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CROSSREFS
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Number of ways of writing 2n+1 as p+q+r where p, q, r are primes with p <= q <= r gives A054860.
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KEYWORD
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AUTHOR
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STATUS
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approved
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