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

Author

Wesley Ivan Hurt, Aug 03 2019

Keywords

• nonn

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    ...
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  n  |     3      4      5      6      7      8      9     10      ...
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a(n) |     0      2      7     11     20     22     40     39      ...
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Links

• Index entries for sequences related to partitions

Crossrefs

Cf. A010051, A309405.

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}]

Formula

a(n) = Sum_{j=1..floor(n/3)} Sum_{i=j..floor((n-j)/2)} (i * A010051(i) + j * A010051(j) + (n-i-j) * A010051(n-i-j)).