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%I #7 May 21 2020 17:28:30
%S 0,0,0,0,0,0,0,0,0,1,3,5,11,17,30,45,72,95,138,183,253,326,433,545,
%T 706,873,1100,1345,1666,2009,2451,2928,3524,4169,4961,5818,6859,7982,
%U 9324,10778,12496,14350,16519,18866,21585,24521,27893,31533,35688,40165
%N Number of prime parts in the partitions of n into 8 parts.
%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F a(n) = Sum_{p=1..floor(n/8)} Sum_{o=p..floor((n-p)/7)} Sum_{m=o..floor((n-o-p)/6)} Sum_{l=m..floor((n-m-o-p)/5)} Sum_{k=l..floor((n-l-m-o-p)/4)} Sum_{j=k..floor((n-k-l-m-o-p)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p)/2)} (A010051(i) + A010051(j) + A010051(k) + A010051(l) + A010051(m) + A010051(o) + A010051(p) + A010051(n-i-j-k-l-m-o-p)).
%t Table[Sum[Sum[Sum[Sum[Sum[Sum[Sum[(PrimePi[i] - PrimePi[i - 1]) + (PrimePi[j] - PrimePi[j - 1]) + (PrimePi[k] - PrimePi[k - 1]) + (PrimePi[l] - PrimePi[l - 1]) + (PrimePi[m] - PrimePi[m - 1]) + (PrimePi[o] - PrimePi[o - 1]) + (PrimePi[p] - PrimePi[p - 1]) + (PrimePi[n - i - j - k - l - m - o - p] - PrimePi[n - i - j - k - l - m - o - p - 1]), {i, j, Floor[(n - j - k - l - m - o - p)/2]}], {j, k, Floor[(n - k - l - m - o - p)/3]}], {k, l, Floor[(n - l - m - o - p)/4]}], {l, m, Floor[(n - m - o - p)/5]}], {m, o, Floor[(n - o - p)/6]}], {o, p, Floor[(n - p)/7]}], {p, Floor[n/8]}], {n, 0, 50}]
%t Table[Count[Flatten[IntegerPartitions[n,{8}]],_?PrimeQ],{n,0,50}] (* _Harvey P. Dale_, May 21 2020 *)
%Y Cf. A010051, A259158, A309436.
%K nonn
%O 0,11
%A _Wesley Ivan Hurt_, Aug 03 2019
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