A236762 Number of partitions of 3n into 3 parts with the middle part prime.

0, 2, 5, 7, 11, 14, 17, 19, 23, 29, 35, 40, 47, 53, 59, 67, 76, 82, 88, 93, 100, 109, 118, 124, 131, 140, 149, 160, 173, 185, 197, 208, 220, 232, 244, 258, 273, 285, 297, 311, 327, 342, 357, 369, 382, 397, 412, 426, 442, 460, 478, 496, 515, 533, 551, 571

Offset

1,2

Links

Table of n, a(n) for n=1..56.

Index entries for sequences related to partitions

Formula

a(n) = Sum_{i=1..n} i * A010051(i) + Sum_{i=1..floor((n - 1)/2)} A010051(n + i) * (n - 2i).

Example

Count the primes in the second columns 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           2           5           7          11      ..   a(n)

Maple

with(numtheory); A236762:=n->sum( i * (pi(i) - pi(i - 1)), i = 1..n) +
sum( (pi(n + i) - pi(n + i - 1)) * (n - 2*i), i = 1..floor((n - 1)/2) ); seq(A236762(n), n=1..100);

Mathematica

Table[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]}], {n, 100}]

Prog

(Sage) def a(n): return sum(1 for L in Partitions(3*n, length=3).list() if is_prime(L[1])) # Ralf Stephan, Feb 03 2014

Crossrefs

Cf. A019298, A235988, A236364, A236758, A010051.

Sequence in context: A079379 A287004 A095272 * A327952 A099835 A133436

Adjacent sequences: A236759 A236760 A236761 * A236763 A236764 A236765

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Jan 30 2014

Status

approved