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
Examples
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
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0 2 4 4 8 .. a(n)
Links
Programs
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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 *)
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) ).