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%I #17 Jan 07 2022 09:02:26
%S 0,0,0,0,0,2,7,11,28,31,56,68,101,117,165,187,267,307,385,445,563,621,
%T 780,878,1044,1181,1405,1545,1828,2019,2298,2535,2901,3141,3588,3915,
%U 4371,4768,5311,5711,6393,6880,7552,8146,8957,9543,10493,11218,12194
%N Sum of the prime parts in the partitions of n into 4 parts.
%H David A. Corneth, <a href="/A309465/b309465.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_{k=1..floor(n/4)} Sum_{j=k..floor((n-k)/3)} Sum_{i=j..floor((n-j-k)/2)} (i * c(i) + j * c(j) + k * c(k) + (n-i-j-k) * c(n-i-j-k)), where c is the prime characteristic (A010051).
%e Figure 1: The partitions of n into 4 parts for n = 8, 9, ..
%e 1+1+1+9
%e 1+1+2+8
%e 1+1+3+7
%e 1+1+4+6
%e 1+1+1+8 1+1+5+5
%e 1+1+2+7 1+2+2+7
%e 1+1+1+7 1+1+3+6 1+2+3+6
%e 1+1+2+6 1+1+4+5 1+2+4+5
%e 1+1+3+5 1+2+2+6 1+3+3+5
%e 1+1+1+6 1+1+4+4 1+2+3+5 1+3+4+4
%e 1+1+1+5 1+1+2+5 1+2+2+5 1+2+4+4 2+2+2+6
%e 1+1+2+4 1+1+3+4 1+2+3+4 1+3+3+4 2+2+3+5
%e 1+1+3+3 1+2+2+4 1+3+3+3 2+2+2+5 2+2+4+4
%e 1+2+2+3 1+2+3+3 2+2+2+4 2+2+3+4 2+3+3+4
%e 2+2+2+2 2+2+2+3 2+2+3+3 2+3+3+3 3+3+3+3
%e --------------------------------------------------------------------------
%e n | 8 9 10 11 12 ...
%e --------------------------------------------------------------------------
%e a(n) | 28 31 56 68 101 ...
%e --------------------------------------------------------------------------
%e - _Wesley Ivan Hurt_, Sep 08 2019
%t Table[Sum[Sum[Sum[i (PrimePi[i] - PrimePi[i - 1]) + j (PrimePi[j] - PrimePi[j - 1]) + k (PrimePi[k] - PrimePi[k - 1]) + (n - i - j - k) (PrimePi[n - i - j - k] - PrimePi[n - i - j - k - 1]), {i, j, Floor[(n - j - k)/2]}], {j, k, Floor[(n - k)/3]}], {k, Floor[n/4]}], {n, 0, 80}]
%Y Cf. A010051, A309465, A309466, A309467, A309468, A309469, A309470, A309471.
%K nonn
%O 0,6
%A _Wesley Ivan Hurt_, Aug 03 2019
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