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A278700
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Number of Goldbach partitions (p,q) of 2n such that all primes from p to q (inclusive) appear as a part in some Goldbach partition of p+q = 2n.
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14
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0, 0, 1, 1, 2, 1, 1, 0, 2, 0, 1, 3, 1, 0, 3, 0, 1, 2, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 5, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 3, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 1, 2, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 0, 1, 0, 0, 1
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OFFSET
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1,5
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COMMENTS
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Records are a(3) = 1, a(5) = 2, a(12) = 3, a(30) = 5, a(165) = 6, a(8021811) = 7. - Charles R Greathouse IV, Nov 30 2016
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LINKS
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FORMULA
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a(n) = Sum_{i=3..n} (c(i) * c(2*n-i) * Product_{k=i..n} (1 - abs(c(k) - c(2*n-k)))), where c is the prime characteristic (A010051).
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EXAMPLE
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a(5) = 2; There are 2 Goldbach partitions of 2*5 = 10, namely (3,7) and (5,5). (3,7) satisfies the condition that every prime from 3 to 7 (inclusive) appears as a part in some Goldbach partition of 10. This is also true of (5,5) since 5 appears in its own partition.
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MATHEMATICA
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Table[Sum[Times @@ Map[Boole@ PrimeQ@ # &, {i, 2 n - i}] * Product[(1 - Abs[Subtract @@ Map[Boole@ PrimeQ@ # &, {k, 2 n - k}]]), {k, i, n}], {i, 3, n}], {n, 89}] (* Michael De Vlieger, Nov 30 2016 *)
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PROG
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(PARI) a(n) = sum(i=3, n, (ispseudoprime(i) * ispseudoprime(2*n-i) * prod(k=i, n, (1-abs(ispseudoprime(k)-ispseudoprime(2*n-k)))))) \\ Felix Fröhlich, Nov 28 2016
(PARI) a(n) = if(n<3, return(0)); my(s, p=n, N=2*n); forprime(q=n, N, while(p<q, if(isprime(N-p), return(s)); p++); p++; if(!isprime(N-q), break); s++); s \\ Charles R Greathouse IV, Nov 30 2016
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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