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 A228553 Sum of the products formed by multiplying together the smaller and larger parts of each Goldbach partition of 2n. 4
 0, 4, 9, 15, 46, 35, 82, 94, 142, 142, 263, 357, 371, 302, 591, 334, 780, 980, 578, 821, 1340, 785, 1356, 1987, 1512, 1353, 2677, 1421, 2320, 4242, 1955, 2803, 4362, 1574, 4021, 5298, 4177, 4159, 6731, 4132, 5593, 9808 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Since the product of each prime pair is semiprime and since we are adding A045917(n) of these, a(n) is expressible as the sum of exactly A045917(n) distinct semiprimes. LINKS Robert Israel, Table of n, a(n) for n = 1..10000 Index entries for sequences related to Goldbach conjecture Index entries for sequences related to partitions FORMULA a(n) = Sum_{i=2..n} c(i) * c(2*n-i) * i * (2*n-i), where c = A010051. a(n) = Sum_{k=(n^2-n+2)/2..(n^2+n-2)/2} c(A105020(k)) * A105020(k), where c = A064911. - Wesley Ivan Hurt, Sep 19 2021 EXAMPLE a(5) = 46. 2*5 = 10 has two Goldbach partitions: (7,3) and (5,5). Taking the products of the larger and smaller parts of these partitions and adding, we get 7*3 + 5*5 = 46. MAPLE with(numtheory); seq(sum( (2*k*i-i^2) * (pi(i)-pi(i-1)) * (pi(2*k-i)-pi(2*k-i-1)), i=2..k), k=1..70); # Alternative: f:= proc(n) local S; S:= select(t -> isprime(t) and isprime(2*n-t), [seq(i, i=3..n, 2)]); add(t*(2*n-t), t=S) end proc: f(2):= 4: map(f, [\$1..200]); # Robert Israel, Nov 29 2020 MATHEMATICA c[n_] := Boole[PrimeQ[n]]; a[n_] := Sum[c[i]*c[2n-i]*i*(2n-i), {i, 2, n}]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Feb 02 2023 *) CROSSREFS Cf. A010051, A045917, A064911, A105020, A185297, A187129. Sequence in context: A301254 A291318 A178379 * A356928 A357807 A337568 Adjacent sequences: A228550 A228551 A228552 * A228554 A228555 A228556 KEYWORD nonn AUTHOR Wesley Ivan Hurt, Aug 25 2013 STATUS approved

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Last modified August 4 13:26 EDT 2024. Contains 374921 sequences. (Running on oeis4.)