%I #6 Jun 25 2022 01:21:01

%S 0,0,0,0,0,0,2,3,3,8,5,12,12,12,7,23,18,38,24,31,24,59,30,73,47,71,49,

%T 113,55,115,40,102,59,171,48,168,100,191,102,220,50,265,89,246,120,

%U 322,109,383,136,348,181,477,158,516,117,468,199,605,133,574,170,600,252,751,133

%N Sum of the largest parts of the partitions of n into exactly 3 prime parts.

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

%F a(n) = Sum_{j=1..floor(n/3)} Sum_{i=j..floor((n-j)/2)} c(i) * c(j) * c(n-i-j) * (n-i-j), where c = A010051.

%F a(n) = A355199(n) - A355197(n) - A355198(n).

%e a(9) = 8; since 9 can be written as the sum of 3 primes in two ways: 2+2+5 = 3+3+3 and the sum of the largest parts of these partitions is 5+3 = 8, we have a(9) = 8.

%t Table[Sum[Sum[(n - i - j) (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 Cf. A010051, A068307, A355197, A355198, A355199.

%K nonn

%O 0,7

%A _Wesley Ivan Hurt_, Jun 23 2022