A237264 Number of partitions of 3n into 3 parts with largest part prime.

0, 2, 4, 4, 8, 7, 13, 15, 22, 21, 28, 29, 36, 35, 44, 45, 54, 55, 67, 70, 83, 84, 96, 99, 116, 119, 135, 138, 154, 154, 170, 172, 187, 189, 208, 211, 231, 235, 259, 264, 285, 286, 306, 310, 334, 337, 361, 366, 389, 390, 413, 416, 441, 443, 468, 471, 496, 498

Offset

1,2

Links

Wesley Ivan Hurt, Table of n, a(n) for n = 1..58

Index entries for sequences related to partitions

Formula

a(n) = Sum_{j=0..n-2} ( Sum_{i=n + 1 + floor(j/2) - floor(1/(j + 1))..n + 2(j + 1)} A010051(i) ).

Example

Count the primes in the first 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           2           4           4           8       ..  a(n)

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}], {n, 50}]
Table[Count[IntegerPartitions[3 n, {3}], _?(PrimeQ[#[[1]]]&)], {n, 60}] (* Harvey P. Dale, Mar 06 2022 *)

Crossrefs

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

Sequence in context: A235999 A093820 A095400 * A265560 A265544 A290221

Adjacent sequences: A237261 A237262 A237263 * A237265 A237266 A237267

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Feb 10 2014

Status

approved