A279536 -id:A279536 - cp's OEIS frontend

A279585 Partial sums of A279536.

0, 0, 1, 3, 8, 11, 12, 12, 21, 21, 22, 41, 42, 42, 63, 63, 64, 74, 75, 75, 79, 79, 80, 80, 80, 83, 83, 83, 84, 152, 153, 153, 153, 158, 158, 158, 159, 159, 163, 163, 164, 189, 190, 190, 193, 193, 194, 194, 194, 197, 197, 197, 205, 205, 205, 210, 210, 210, 211, 223

Offset: 1

Wesley Ivan Hurt, Dec 15 2016

Eric Weisstein's World of Mathematics, Goldbach Partition
Wikipedia, Goldbach's conjecture
Index entries for sequences related to Goldbach conjecture
Index entries for sequences related to partitions

Cf. A010051, A279536.

with(numtheory): t:=n->add(add( (pi(i)-pi(i-1)) * (pi(2*h-i)-pi(2*h-i-1)) * add(mobius(j)^2, j=i..2*h-i) * (product(1-abs((pi(k)-pi(k-1))-(pi(2*h-k)-pi(2*h-k-1))), k=i..h)), i=3..h), h=1..n): seq(t(n), n=1..60);

a(n) = Sum_{h=1..n} ( Sum_{i=3..h} A010051(i) * A010051(2h-i) * ( Sum_{j=i..2h-i} mu(j)^2 ) * (Product_{k=i..h} (1-abs(A010051(k)-A010051(2h-k))))).

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.

Original entry on oeis.org

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

Offset: 1

Wesley Ivan Hurt, Dec 13 2016

Eric Weisstein's World of Mathematics, Goldbach Partition
Wikipedia, Goldbach's conjecture
Index entries for sequences related to Goldbach conjecture
Index entries for sequences related to partitions

Cf. A000720, A010051, A278700, A279481, A279536.

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);

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 *)

a(n) = Sum_{i=3..n} c(i) * c(2*n-i) * (pi(2*n-i)-pi(i-1)) * (Product_{k=i..n} (1-abs(c(k)-c(2*n-k)))), where pi is the prime counting function (A000720), and c is the prime characteristic (A010051).
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)

cp's OEIS Frontend

A279585 Partial sums of A279536.

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