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A243485
Sum of all the products formed by multiplying the corresponding smaller and larger parts of the Goldbach partitions of n.
3
0, 0, 0, 4, 6, 9, 10, 15, 14, 46, 0, 35, 22, 82, 26, 94, 0, 142, 34, 142, 38, 263, 0, 357, 46, 371, 0, 302, 0, 591, 58, 334, 62, 780, 0, 980, 0, 578, 74, 821, 0, 1340, 82, 785, 86, 1356, 0, 1987, 94, 1512, 0, 1353, 0, 2677, 106, 1421, 0, 2320, 0, 4242, 118
OFFSET
1,4
COMMENTS
a(n) is even for odd n.
If Goldbach's conjecture is true, a(n) > 0 for all even n > 2.
Sum of the areas of the distinct rectangles with prime length and width such that L + W = n, W <= L. For example, a(16) = 94; the two rectangles are 3 X 13 and 5 X 11, and the sum of their areas is 3*13 + 5*11 = 94. - Wesley Ivan Hurt, Oct 28 2017
FORMULA
a(n) = Sum_{i=2..n/2} i*(n-i) * A064911(i*(n-i)).
a(n) = Sum_{i=1..floor(n/2)} i * (n-i) * A010051(i) * A010051(n-i). - Wesley Ivan Hurt, Oct 29 2017
MAPLE
with(numtheory): A243485:=n->add(i*(n-i)*(pi(i)-pi(i-1))*(pi(n-i)-pi(n-i-1)), i=1..floor(n/2)): seq(A243485(n), n=1..100); # Wesley Ivan Hurt, Oct 29 2017
MATHEMATICA
Table[Sum[i*(n - i)*Floor[2/PrimeOmega[i (n - i)]], {i, 2, n/2}], {n,
50}]
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Wesley Ivan Hurt, Jun 05 2014
STATUS
approved