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A228553
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Sum of the products formed by multiplying together the smaller and larger parts of each Goldbach partition of 2n.
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4
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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
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OFFSET
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1,2
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COMMENTS
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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.
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LINKS
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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
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FORMULA
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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
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EXAMPLE
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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.
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MAPLE
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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
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MATHEMATICA
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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 *)
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CROSSREFS
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Cf. A010051, A045917, A064911, A105020, A185297, A187129.
Sequence in context: A301254 A291318 A178379 * A356928 A357807 A337568
Adjacent sequences: A228550 A228551 A228552 * A228554 A228555 A228556
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KEYWORD
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nonn
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AUTHOR
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Wesley Ivan Hurt, Aug 25 2013
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STATUS
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approved
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