A309405 Number of prime parts in the partitions of n into 3 parts.

0, 0, 0, 0, 1, 3, 5, 7, 8, 12, 12, 16, 17, 21, 22, 29, 29, 34, 35, 41, 42, 50, 50, 58, 59, 67, 68, 77, 78, 86, 87, 96, 97, 108, 108, 119, 120, 130, 131, 144, 144, 155, 156, 168, 169, 182, 183, 197, 198, 212, 213, 228, 228, 242, 243, 258, 259, 275, 275, 291

Offset: 0

Author

Wesley Ivan Hurt, Jul 30 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      1      3      5      7      8     12     12      ...
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Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000
• Index entries for sequences related to partitions

Crossrefs

Cf. A010051, A069905.

Programs

Mathematica

Table[Sum[Sum[(PrimePi[i] - PrimePi[i - 1]) + (PrimePi[j] - PrimePi[j - 1]) + (PrimePi[n - i - j] - PrimePi[n - i - j - 1]), {i, j, Floor[(n - j)/2]}], {j, Floor[n/3]}], {n, 0, 80}]
Table[Count[Flatten[IntegerPartitions[n,{3}]],?PrimeQ],{n,0,60}] (* _Harvey P. Dale, Jun 13 2025 *)

Formula

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