Number of ways in which 2n-1 can be written as the sum of 3 odd composite numbers 7 < u < v < w.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 1, 2, 1, 2, 3, 1, 2, 5, 2, 4, 6, 2, 5, 8, 4, 7, 9, 6, 8, 12, 7, 10, 16, 9, 11, 18, 10, 16, 21, 13, 18, 24, 16, 21, 27, 20, 23, 34, 21, 26, 38, 24, 32, 42, 27, 36, 45, 34, 40, 51, 40

1

Complement of the ternary Goldbach problem; instead of writing an odd number as the sum of three primes, we write it as the sum of distinct odd composite numbers. The following numbers have just one representation: 45, 49, 51, 55, 59, and 65. Odd numbers less than 45 have zero representations.

T. D. Noe, Plot of 1000 terms

T. D. Noe, Table of 1000 terms

Harald Andres Helfgott, The ternary Goldbach problem, arXiv 1501.05438

(Mma) nn = 200; ps = Prime[Range[PrimePi[nn]]]; oddNP = Complement[Range[9, nn, 2], ps]; Table[s = IntegerPartitions[p, {3}, oddNP]; Length[Select[s, Length[Union[#]] == 3 &]], {p, 1, nn, 2}]

Cf. A024681 (primes instead of odd numbers 2n-1), S000457, S000458, S000459.

nonn

T. D. Noe, Jan 24 2015