A236758 Number of partitions of 3*n into 3 parts with smallest part prime.

0, 1, 3, 6, 10, 14, 20, 25, 32, 37, 45, 51, 61, 68, 79, 86, 98, 106, 120, 129, 144, 153, 169, 179, 196, 206, 223, 233, 251, 262, 282, 294, 315, 327, 348, 360, 382, 395, 418, 431, 455, 469, 495, 510, 537, 552, 580, 596, 625, 641, 670, 686, 716, 733, 764, 781

Offset

1,3

Links

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

Index entries for sequences related to partitions

Formula

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

Example

Count the primes in last column 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           1           3           6           10      ..   a(n)

Maple

with(numtheory); A236758:=n->sum((pi(n) - pi(n-1)) * (2*n - 2*i + 1 - floor((n - i + 1)/2)), i=1..n); seq(A236758(n), n=1..100);

Mathematica

Table[Sum[(PrimePi[i] - PrimePi[i - 1]) (2 n - 2 i + 1 - Floor[(n - i + 1)/2]), {i, n}], {n, 100}]

Prog

(Sage) def a(n): return sum(1 for L in Partitions(3*n, length=3).list() if is_prime(L[2]))

Crossrefs

Cf. A019298, A235988, A236364, A236762, A010051 (for function isprime).

Sequence in context: A330257 A079552 A334454 * A272058 A244360 A183863

Adjacent sequences: A236755 A236756 A236757 * A236759 A236760 A236761

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Jan 30 2014

Status

approved