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A237264
Number of partitions of 3n into 3 parts with largest part prime.
3
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
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
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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
KEYWORD
nonn,easy
AUTHOR
Wesley Ivan Hurt, Feb 10 2014
STATUS
approved