A309426 Number of prime parts in the partitions of n into 4 parts.

0, 0, 0, 0, 0, 1, 3, 5, 11, 12, 19, 23, 32, 36, 47, 53, 68, 77, 92, 103, 123, 134, 157, 173, 197, 216, 245, 265, 299, 323, 357, 385, 425, 454, 499, 534, 580, 619, 671, 711, 770, 816, 875, 926, 993, 1044, 1116, 1175, 1249, 1314, 1396, 1462, 1552, 1625, 1714

Offset

0,7

Links

Table of n, a(n) for n=0..54.

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)} (A010051(i) + A010051(j) + A010051(k) + A010051(n-i-j-k)).

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) |     11          12          19          23          32        ...
--------------------------------------------------------------------------
- Wesley Ivan Hurt, Sep 08 2019

Mathematica

Table[Sum[Sum[Sum[(PrimePi[i] - PrimePi[i - 1]) + (PrimePi[j] - PrimePi[j - 1]) + (PrimePi[k] - PrimePi[k - 1]) + (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, 50}]
Table[Count[Flatten[IntegerPartitions[n, {4}]], _?PrimeQ], {n, 0, 60}] (* Harvey P. Dale, Oct 04 2019 *)

Crossrefs

Cf. A010051, A026810.

Sequence in context: A066281 A072063 A242269 * A115398 A014597 A357369

Adjacent sequences: A309423 A309424 A309425 * A309427 A309428 A309429

Keyword

nonn

Author

Wesley Ivan Hurt, Aug 01 2019

Status

approved