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A294345
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Sum of the products of the smaller and larger parts of the Goldbach partitions of n into two distinct parts.
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1
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0, 0, 0, 0, 6, 0, 10, 15, 14, 21, 0, 35, 22, 33, 26, 94, 0, 142, 34, 142, 38, 142, 0, 357, 46, 202, 0, 302, 0, 591, 58, 334, 62, 491, 0, 980, 0, 217, 74, 821, 0, 1340, 82, 785, 86, 827, 0, 1987, 94, 1512, 0, 1353, 0, 2677, 106, 1421, 0, 1479, 0, 4242, 118
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OFFSET
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1,5
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COMMENTS
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Sum of the areas of the distinct rectangles with prime length and width such that L + W = n, W < L. For example, a(14) = 33; the only rectangle is 3 X 11 and 3*11 = 33 (the 7 X 7 rectangle is not considered since we have W < L).
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LINKS
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FORMULA
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a(n) = Sum_{i=2..floor((n-1)/2)} i * (n-i) * c(i) * c(n-i), where c is the prime characteristic (A010051).
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EXAMPLE
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a(22) = 142; the Goldbach partitions of 22 are (19,3), (17,5) and (11,11) (we do not consider (11,11) since we only count prime parts which are distinct). Then the sum of the products of the smaller and larger parts from each partition is 19*3 + 17*5 = 142.
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MAPLE
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with(numtheory): A294345:=n->add(i*(n-i)*(pi(i)-pi(i-1))*(pi(n-i)-pi(n-i-1)), i=2..floor((n-1)/2)): seq(A294345(n), n=1..100);
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MATHEMATICA
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Table[Sum[i (n - i) (PrimePi[i] - PrimePi[i - 1]) (PrimePi[n - i] - PrimePi[n - i - 1]), {i, 2, Floor[(n-1)/2]}], {n, 60}]
Table[Total[Times@@@Select[IntegerPartitions[n, {2}], AllTrue[#, PrimeQ] && #[[1]]!=#[[2]]&]], {n, 70}] (* Harvey P. Dale, Jul 29 2021 *)
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PROG
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(PARI) a(n) = sum(i=1, (n-1)\2, i*isprime(i)*(n-i)*isprime(n-i)); \\ Michel Marcus, Nov 08 2017
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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