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URL: https://oeis.org/A236762

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A236762
Number of partitions of 3n into 3 parts with the middle part prime.
5
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
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
(SageMath) 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
KEYWORD
nonn,easy
AUTHOR
Wesley Ivan Hurt, Jan 30 2014
STATUS
approved