A308724 - OEIS

%I #44 Aug 03 2019 21:47:03

%S 0,0,0,0,2,7,11,20,22,40,39,59,61,87,89,140,137,176,178,234,236,318,

%T 313,399,401,499,501,612,614,712,714,841,843,1012,1003,1178,1180,1338,

%U 1340,1567,1556,1751,1753,1989,1991,2270,2272,2574,2576,2902,2904,3247

%N Sum of the prime parts in the partitions of n into 3 parts.

%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>

%F 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)).

%e Figure 1: The partitions of n into 3 parts for n = 3, 4, ...

%e                                                           1+1+8

%e                                                    1+1+7  1+2+7

%e                                                    1+2+6  1+3+6

%e                                             1+1+6  1+3+5  1+4+5

%e                                      1+1+5  1+2+5  1+4+4  2+2+6

%e                               1+1+4  1+2+4  1+3+4  2+2+5  2+3+5

%e                        1+1+3  1+2+3  1+3+3  2+2+4  2+3+4  2+4+4

%e          1+1+1  1+1+2  1+2+2  2+2+2  2+2+3  2+3+3  3+3+3  3+3+4    ...

%e -----------------------------------------------------------------------

%e   n  |     3      4      5      6      7      8      9     10      ...

%e -----------------------------------------------------------------------

%e a(n) |     0      2      7     11     20     22     40     39      ...

%e -----------------------------------------------------------------------

%t 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}]

%Y Cf. A010051, A309405.

%K nonn

%O 0,5

%A _Wesley Ivan Hurt_, Aug 03 2019