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

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 3, 0, 0, 6, 0, 3, 10, 0, 6, 15, 3, 9, 21, 9, 12, 28, 12, 18, 42, 15, 27, 52, 18, 39, 61, 30, 51, 72, 42, 57, 90, 54, 66, 118, 60, 78, 133, 72, 105, 150, 90, 123, 166, 114, 135, 192, 138, 153, 231, 141, 171, 265

1

This sequence is like S000457, except the numbers u,v,w are orderless. Note that the first odd composite number is 9.

T. D. Noe, Plot of 1000 terms

T. D. Noe, Table of 1000 terms

(Mma) nn = 200; ps = Prime[Range[PrimePi[nn]]]; oddNP = Complement[Range[9, nn, 2], ps]; Table[s = IntegerPartitions[p, {3}, oddNP]; cnt = 0; Do[k = Length[Union[i]]; If[k == 3, cnt = cnt + 6, If[k == 2, cnt = cnt + 3, cnt++]], {i, s}]; cnt, {p, 1, nn, 2}]

Cf. S000457.

nonn

T. D. Noe, Jan 26 2015