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 A279315 Count the primes appearing in each interval [p,q] where (p,q) is a Goldbach partition of 2n such that all primes from p to q (inclusive) appear as a part in some Goldbach partition of p+q = 2n, and then add the results. 6
 0, 0, 1, 2, 4, 2, 1, 0, 6, 0, 1, 12, 1, 0, 12, 0, 1, 6, 1, 0, 2, 0, 1, 0, 0, 2, 0, 0, 1, 30, 1, 0, 0, 2, 0, 0, 1, 0, 2, 0, 1, 12, 1, 0, 2, 0, 1, 0, 0, 2, 0, 0, 4, 0, 0, 2, 0, 0, 1, 6, 1, 0, 0, 2, 0, 0, 1, 0, 2, 0, 1, 2, 1, 0, 0, 2, 0, 0, 1, 0, 6, 0, 1, 0, 0, 2, 0 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 LINKS Eric Weisstein's World of Mathematics, Goldbach Partition Wikipedia, Goldbach's conjecture FORMULA a(n) = Sum_{i=3..n} (A010051(i) * A010051(2n-i) * (pi(2*n-i)-pi(i-1)) * (Product_{k=i..n} (1-abs(A010051(k)-A010051(2n-k)))), where pi is the prime counting function (A000720). From Wesley Ivan Hurt, Dec 17 2016: (Start) a(n) = A010051(n)*A278700(n)^2+(1-A010051(n))*A278700(n)*(A278700(n)+1). a(n) <= A279536(n). (End) MAPLE with(numtheory): A279315:=n->add( (pi(i)-pi(i-1)) * (pi(2*n-i)-pi(2*n-i-1)) * (pi(2*n-i)-pi(i-1)) * (product(1-abs((pi(k)-pi(k-1))-(pi(2*n-k)-pi(2*n-k-1))), k=i..n)), i=3..n): seq(A279315(n), n=1..100); MATHEMATICA f[n_] := Sum[ Boole[PrimeQ[i]] Boole[PrimeQ[ 2n -i]] (PrimePi[ 2n -i] - PrimePi[i -1]) Product[(1 - Abs[Boole[PrimeQ[k]] - Boole[PrimeQ[ 2n -k]]]), {k, i, n}], {i, 3, n}]; Array[f, 80] (* Robert G. Wilson v, Dec 15 2016 *) CROSSREFS Cf. A000720, A010051, A278700, A279481, A279536. Sequence in context: A273240 A201316 A105023 * A303293 A201558 A052285 Adjacent sequences:  A279312 A279313 A279314 * A279316 A279317 A279318 KEYWORD nonn,easy AUTHOR Wesley Ivan Hurt, Dec 13 2016 STATUS approved

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Last modified June 1 18:43 EDT 2020. Contains 334762 sequences. (Running on oeis4.)