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
Examples
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
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0 1 3 6 10 .. a(n)
Programs
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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);
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Mathematica
Table[Sum[(PrimePi[i] - PrimePi[i - 1]) (2 n - 2 i + 1 - Floor[(n - i + 1)/2]), {i, n}], {n, 100}] -
Sage
def a(n): return sum(1 for L in Partitions(3*n,length=3).list() if is_prime(L[2]))
Formula
a(n) = Sum_{i=1..n} A010051(i) * (2*n - 2*i + 1 - floor((n - i + 1)/2)).