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%I A355199 #4 Jun 25 2022 01:21:25
%S A355199 0,0,0,0,0,0,6,7,8,18,10,22,24,26,14,45,32,68,36,57,40,105,44,115,72,
%T A355199 125,78,189,84,203,60,186,96,297,68,280,144,333,152,390,80,451,126,
%U A355199 430,176,540,138,611,192,588,250,765,208,848,162,770,280,969,174,944,240,976,372
%N A355199 Sum of all the parts in the partitions of n into exactly 3 prime parts.
%H A355199 <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F A355199 a(n) = 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.
%F A355199 a(n) = n * A068307(n) for n > 0.
%F A355199 a(n) = A355196(n) + A355197(n) + A355198(n).
%e A355199 a(9) = 18; since 9 can be written as the sum of 3 primes in two ways: 2+2+5 = 3+3+3 and the sum of all the parts of these partitions is (2+2+5)+(3+3+3) = 18, we have a(9) = 18.
%t A355199 Table[n*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}]
%Y A355199 Cf. A010051, A068307, A355196, A355197, A355198.
%K A355199 nonn
%O A355199 0,7
%A A355199 _Wesley Ivan Hurt_, Jun 23 2022

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