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

%I #35 Mar 06 2022 13:09:11

%S 0,2,4,4,8,7,13,15,22,21,28,29,36,35,44,45,54,55,67,70,83,84,96,99,

%T 116,119,135,138,154,154,170,172,187,189,208,211,231,235,259,264,285,

%U 286,306,310,334,337,361,366,389,390,413,416,441,443,468,471,496,498

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

%H Wesley Ivan Hurt, <a href="/A237264/b237264.txt">Table of n, a(n) for n = 1..58</a>

%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>

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

%e Count the primes in the first column for a(n).

%e 13 + 1 + 1

%e 12 + 2 + 1

%e 11 + 3 + 1

%e 10 + 4 + 1

%e 9 + 5 + 1

%e 8 + 6 + 1

%e 7 + 7 + 1

%e 10 + 1 + 1 11 + 2 + 2

%e 9 + 2 + 1 10 + 3 + 2

%e 8 + 3 + 1 9 + 4 + 2

%e 7 + 4 + 1 8 + 5 + 2

%e 6 + 5 + 1 7 + 6 + 2

%e 7 + 1 + 1 8 + 2 + 2 9 + 3 + 3

%e 6 + 2 + 1 7 + 3 + 2 8 + 4 + 3

%e 5 + 3 + 1 6 + 4 + 2 7 + 5 + 3

%e 4 + 4 + 1 5 + 5 + 2 6 + 6 + 3

%e 4 + 1 + 1 5 + 2 + 2 6 + 3 + 3 7 + 4 + 4

%e 3 + 2 + 1 4 + 3 + 2 5 + 4 + 3 6 + 5 + 4

%e 1 + 1 + 1 2 + 2 + 2 3 + 3 + 3 4 + 4 + 4 5 + 5 + 5

%e 3(1) 3(2) 3(3) 3(4) 3(5) .. 3n

%e ---------------------------------------------------------------------

%e 0 2 4 4 8 .. a(n)

%t 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}]

%t Table[Count[IntegerPartitions[3 n,{3}],_?(PrimeQ[#[[1]]]&)],{n,60}] (* _Harvey P. Dale_, Mar 06 2022 *)

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

%K nonn,easy

%O 1,2

%A _Wesley Ivan Hurt_, Feb 10 2014