This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A309405 #28 Jun 19 2025 22:01:20
%S A309405 0,0,0,0,1,3,5,7,8,12,12,16,17,21,22,29,29,34,35,41,42,50,50,58,59,67,
%T A309405 68,77,78,86,87,96,97,108,108,119,120,130,131,144,144,155,156,168,169,
%U A309405 182,183,197,198,212,213,228,228,242,243,258,259,275,275,291
%N A309405 Number of prime parts in the partitions of n into 3 parts.
%H A309405 Harvey P. Dale, <a href="/A309405/b309405.txt">Table of n, a(n) for n = 0..1000</a>
%H A309405 <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F A309405 a(n) = Sum_{j=1..floor(n/3)} Sum_{i=j..floor((n-j)/2)} (c(i) + c(j) + c(n-i-j)), where c = A010051.
%e A309405 Figure 1: The partitions of n into 3 parts for n = 3, 4, ...
%e A309405 1+1+8
%e A309405 1+1+7 1+2+7
%e A309405 1+2+6 1+3+6
%e A309405 1+1+6 1+3+5 1+4+5
%e A309405 1+1+5 1+2+5 1+4+4 2+2+6
%e A309405 1+1+4 1+2+4 1+3+4 2+2+5 2+3+5
%e A309405 1+1+3 1+2+3 1+3+3 2+2+4 2+3+4 2+4+4
%e A309405 1+1+1 1+1+2 1+2+2 2+2+2 2+2+3 2+3+3 3+3+3 3+3+4 ...
%e A309405 -----------------------------------------------------------------------
%e A309405 n | 3 4 5 6 7 8 9 10 ...
%e A309405 -----------------------------------------------------------------------
%e A309405 a(n) | 0 1 3 5 7 8 12 12 ...
%e A309405 -----------------------------------------------------------------------
%t A309405 Table[Sum[Sum[(PrimePi[i] - PrimePi[i - 1]) + (PrimePi[j] - PrimePi[j - 1]) + (PrimePi[n - i - j] - PrimePi[n - i - j - 1]), {i, j, Floor[(n - j)/2]}], {j, Floor[n/3]}], {n, 0, 80}]
%t A309405 Table[Count[Flatten[IntegerPartitions[n,{3}]],_?PrimeQ],{n,0,60}] (* _Harvey P. Dale_, Jun 13 2025 *)
%Y A309405 Cf. A010051, A069905.
%K A309405 nonn
%O A309405 0,6
%A A309405 _Wesley Ivan Hurt_, Jul 30 2019