

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
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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



FORMULA

a(n) = Sum_{i=2..n} c(i) * c(2*ni) * i * (2*ni), where c = A010051.


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*ii^2) * (pi(i)pi(i1)) * (pi(2*ki)pi(2*ki1)), i=2..k), k=1..70);
# Alternative:
f:= proc(n)
local S;
S:= select(t > isprime(t) and isprime(2*nt), [seq(i, i=3..n, 2)]);
add(t*(2*nt), t=S)
end proc:
f(2):= 4:


MATHEMATICA

c[n_] := Boole[PrimeQ[n]];
a[n_] := Sum[c[i]*c[2ni]*i*(2ni), {i, 2, n}];


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



