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%I #24 Jan 07 2022 09:02:16
%S 0,0,0,0,0,0,2,7,11,28,41,67,88,136,169,248,295,413,496,652,772,1001,
%T 1161,1469,1697,2096,2398,2923,3316,3975,4501,5302,5955,6953,7757,
%U 8994,9988,11450,12674,14427,15883,17992,19741,22176,24268,27149,29569,32919
%N Sum of the prime parts in the partitions of n into 5 parts.
%H David A. Corneth, <a href="/A309466/b309466.txt">Table of n, a(n) for n = 0..9999</a>
%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F a(n) = Sum_{l=1..floor(n/5)} Sum_{k=l..floor((n-1)/4)} Sum_{j=k..floor((n-k-l)/3)} Sum_{i=j..floor((n-j-k-l)/2)} (i * c(i) + j * c(j) + k * c(k) + l * c(l) + (n-i-j-k-l) * c(n-i-j-k-l)), where c is the prime characteristic (A010051).
%e The partitions of n into 5 parts for n = 10, 11, ..
%e 1+1+1+1+10
%e 1+1+1+2+9
%e 1+1+1+3+8
%e 1+1+1+4+7
%e 1+1+1+5+6
%e 1+1+1+1+9 1+1+2+2+8
%e 1+1+1+2+8 1+1+2+3+7
%e 1+1+1+3+7 1+1+2+4+6
%e 1+1+1+4+6 1+1+2+5+5
%e 1+1+1+5+5 1+1+3+3+6
%e 1+1+1+1+8 1+1+2+2+7 1+1+3+4+5
%e 1+1+1+2+7 1+1+2+3+6 1+1+4+4+4
%e 1+1+1+3+6 1+1+2+4+5 1+2+2+2+7
%e 1+1+1+1+7 1+1+1+4+5 1+1+3+3+5 1+2+2+3+6
%e 1+1+1+2+6 1+1+2+2+6 1+1+3+4+4 1+2+2+4+5
%e 1+1+1+3+5 1+1+2+3+5 1+2+2+2+6 1+2+3+3+5
%e 1+1+1+1+6 1+1+1+4+4 1+1+2+4+4 1+2+2+3+5 1+2+3+4+4
%e 1+1+1+2+5 1+1+2+2+5 1+1+3+3+4 1+2+2+4+4 1+3+3+3+4
%e 1+1+1+3+4 1+1+2+3+4 1+2+2+2+5 1+2+3+3+4 2+2+2+2+6
%e 1+1+2+2+4 1+1+3+3+3 1+2+2+3+4 1+3+3+3+3 2+2+2+3+5
%e 1+1+2+3+3 1+2+2+2+4 1+2+3+3+3 2+2+2+2+5 2+2+2+4+4
%e 1+2+2+2+3 1+2+2+3+3 2+2+2+2+4 2+2+2+3+4 2+2+3+3+4
%e 2+2+2+2+2 2+2+2+2+3 2+2+2+3+3 2+2+3+3+3 2+3+3+3+3
%e --------------------------------------------------------------------------
%e n | 10 11 12 13 14 ...
%e --------------------------------------------------------------------------
%e a(n) | 41 67 88 136 169 ...
%e --------------------------------------------------------------------------
%e - _Wesley Ivan Hurt_, Sep 12 2019
%t Table[Total[Select[Flatten[IntegerPartitions[n,{5}]],PrimeQ]],{n,0,50}] (* _Harvey P. Dale_, Dec 31 2021 *)
%Y Cf. A010051, A309427, A309465, A309466, A309467, A309468, A309469, A309470, A309471.
%K nonn
%O 0,7
%A _Wesley Ivan Hurt_, Aug 03 2019