A237669 Number of prime parts in the partitions of 3n into 3 parts.

0, 5, 12, 17, 29, 35, 50, 59, 77, 87, 108, 120, 144, 156, 182, 198, 228, 243, 275, 292, 327, 346, 383, 402, 443, 465, 507, 531, 578, 601, 649, 674, 722, 748, 800, 829, 886, 915, 974, 1006, 1067, 1097, 1158, 1189, 1253, 1286, 1353, 1388, 1456, 1491, 1561

Offset: 1

Author

Wesley Ivan Hurt, Feb 11 2014

Keywords

• nonn
• easy

Examples

			Count the primes in the partitions of 3n into 3 parts for a(n).
                                               13 + 1 + 1
                                               12 + 2 + 1
                                               11 + 3 + 1
                                               10 + 4 + 1
                                                9 + 5 + 1
                                                8 + 6 + 1
                                                7 + 7 + 1
                                   10 + 1 + 1  11 + 2 + 2
                                    9 + 2 + 1  10 + 3 + 2
                                    8 + 3 + 1   9 + 4 + 2
                                    7 + 4 + 1   8 + 5 + 2
                                    6 + 5 + 1   7 + 6 + 2
                        7 + 1 + 1   8 + 2 + 2   9 + 3 + 3
                        6 + 2 + 1   7 + 3 + 2   8 + 4 + 3
                        5 + 3 + 1   6 + 4 + 2   7 + 5 + 3
                        4 + 4 + 1   5 + 5 + 2   6 + 6 + 3
            4 + 1 + 1   5 + 2 + 2   6 + 3 + 3   7 + 4 + 4
            3 + 2 + 1   4 + 3 + 2   5 + 4 + 3   6 + 5 + 4
1 + 1 + 1   2 + 2 + 2   3 + 3 + 3   4 + 4 + 4   5 + 5 + 5
   3(1)        3(2)        3(3)        3(4)        3(5)     ..   3n
---------------------------------------------------------------------
    0           5           12          17          29      ..  a(n)
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..300
• Index entries for sequences related to partitions

Crossrefs

Cf. A010051, A019298, A236364, A236370, A236758, A236762, A237264.

Programs

Mathematica

Table[Sum[Sum[PrimePi[i] - PrimePi[i - 1], {i, n + Floor[j/2] + 1 - Floor[1/(j + 1)], n + 2 (j + 1)}], {j, 0, n - 2}] + Sum[i (PrimePi[i] - PrimePi[i - 1]), {i, n}] + Sum[(PrimePi[n + i] - PrimePi[n + i - 1]) (n - 2 i), {i, Floor[(n - 1)/2]}] + Sum[(PrimePi[i] - PrimePi[i - 1]) (2 n - 2 i + 1 - Floor[(n - i + 1)/2]), {i, n}], {n, 70}]
Table[Count[Flatten[IntegerPartitions[3 n,{3}]],?PrimeQ],{n,60}] (* _Harvey P. Dale, Oct 16 2016 *)

Formula

a(n) = A237264(n) + A236762(n) + A236758(n).