A308724 Sum of the prime parts in the partitions of n into 3 parts.
0, 0, 0, 0, 2, 7, 11, 20, 22, 40, 39, 59, 61, 87, 89, 140, 137, 176, 178, 234, 236, 318, 313, 399, 401, 499, 501, 612, 614, 712, 714, 841, 843, 1012, 1003, 1178, 1180, 1338, 1340, 1567, 1556, 1751, 1753, 1989, 1991, 2270, 2272, 2574, 2576, 2902, 2904, 3247
Offset: 0
Keywords
Examples
Figure 1: The partitions of n into 3 parts for n = 3, 4, ...
1+1+8
1+1+7 1+2+7
1+2+6 1+3+6
1+1+6 1+3+5 1+4+5
1+1+5 1+2+5 1+4+4 2+2+6
1+1+4 1+2+4 1+3+4 2+2+5 2+3+5
1+1+3 1+2+3 1+3+3 2+2+4 2+3+4 2+4+4
1+1+1 1+1+2 1+2+2 2+2+2 2+2+3 2+3+3 3+3+3 3+3+4 ...
-----------------------------------------------------------------------
n | 3 4 5 6 7 8 9 10 ...
-----------------------------------------------------------------------
a(n) | 0 2 7 11 20 22 40 39 ...
-----------------------------------------------------------------------
Programs
-
Mathematica
Table[Sum[Sum[i (PrimePi[i] - PrimePi[i - 1]) + j (PrimePi[j] - PrimePi[j - 1]) + (n - i - j) (PrimePi[n - i - j] - PrimePi[n - i - j - 1]), {i, j, Floor[(n - j)/2]}], {j, Floor[n/3]}], {n, 0, 50}]