A309465 Sum of the prime parts in the partitions of n into 4 parts.

0, 0, 0, 0, 0, 2, 7, 11, 28, 31, 56, 68, 101, 117, 165, 187, 267, 307, 385, 445, 563, 621, 780, 878, 1044, 1181, 1405, 1545, 1828, 2019, 2298, 2535, 2901, 3141, 3588, 3915, 4371, 4768, 5311, 5711, 6393, 6880, 7552, 8146, 8957, 9543, 10493, 11218, 12194

Offset

0,6

Links

David A. Corneth, Table of n, a(n) for n = 0..9999

Index entries for sequences related to partitions

Formula

a(n) = Sum_{k=1..floor(n/4)} Sum_{j=k..floor((n-k)/3)} Sum_{i=j..floor((n-j-k)/2)} (i * c(i) + j * c(j) + k * c(k) + (n-i-j-k) * c(n-i-j-k)), where c is the prime characteristic (A010051).

Example

Figure 1: The partitions of n into 4 parts for n = 8, 9, ..
                                                         1+1+1+9
                                                         1+1+2+8
                                                         1+1+3+7
                                                         1+1+4+6
                                             1+1+1+8     1+1+5+5
                                             1+1+2+7     1+2+2+7
                                 1+1+1+7     1+1+3+6     1+2+3+6
                                 1+1+2+6     1+1+4+5     1+2+4+5
                                 1+1+3+5     1+2+2+6     1+3+3+5
                     1+1+1+6     1+1+4+4     1+2+3+5     1+3+4+4
         1+1+1+5     1+1+2+5     1+2+2+5     1+2+4+4     2+2+2+6
         1+1+2+4     1+1+3+4     1+2+3+4     1+3+3+4     2+2+3+5
         1+1+3+3     1+2+2+4     1+3+3+3     2+2+2+5     2+2+4+4
         1+2+2+3     1+2+3+3     2+2+2+4     2+2+3+4     2+3+3+4
         2+2+2+2     2+2+2+3     2+2+3+3     2+3+3+3     3+3+3+3
--------------------------------------------------------------------------
  n  |      8           9          10          11          12        ...
--------------------------------------------------------------------------
a(n) |     28          31          56          68         101        ...
--------------------------------------------------------------------------
- Wesley Ivan Hurt, Sep 08 2019

Mathematica

Table[Sum[Sum[Sum[i (PrimePi[i] - PrimePi[i - 1]) + j (PrimePi[j] - PrimePi[j - 1]) + k (PrimePi[k] - PrimePi[k - 1]) + (n - i - j - k) (PrimePi[n - i - j - k] - PrimePi[n - i - j - k - 1]), {i, j, Floor[(n - j - k)/2]}], {j, k, Floor[(n - k)/3]}], {k, Floor[n/4]}], {n, 0, 80}]

Crossrefs

Cf. A010051, A309465, A309466, A309467, A309468, A309469, A309470, A309471.

Sequence in context: A024481 A024591 A350711 * A309466 A309467 A309468

Adjacent sequences: A309462 A309463 A309464 * A309466 A309467 A309468

Keyword

nonn

Author

Wesley Ivan Hurt, Aug 03 2019

Status

approved