A308925 Sum of the largest parts in the partitions of n into 6 primes.
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 3, 3, 8, 8, 15, 20, 17, 24, 35, 42, 50, 66, 61, 92, 102, 122, 129, 180, 150, 237, 233, 296, 260, 370, 300, 463, 398, 521, 467, 708, 527, 845, 667, 935, 768, 1158, 839, 1372, 1039, 1547, 1233, 1898, 1294, 2217, 1612
Offset: 0
Keywords
Links
Programs
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Maple
N:= proc(m,k,n) option remember; local q,t; if m = 1 then if k=n and isprime(k) then return 1 else return 0 fi fi; if m*k < n then return 0 fi; t:= 0; q:= ceil((n-k)/(m-1))-1; do q:= nextprime(q); if q > min(k, n-k) then return t fi; t:= t + procname(m-1,q,n-k) od; end proc: F:= proc(n) local p, q, t; p:= ceil(n/6)-1; t:= 0; do p:= nextprime(p); if p >= n then return t fi; q:= ceil((n-p)/5)-1; do q:= nextprime(q); if q > min(p,n-p) then break fi; t:= t + p*N(5,q,n-p); od od end proc: map(F, [$0..100]); # Robert Israel, Jul 02 2019 -
Mathematica
Table[Sum[Sum[Sum[Sum[Sum[(n - i - j - k - l - m)*(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[n - i - j - k - l - m] - PrimePi[n - i - j - k - l - m - 1]), {i, j, Floor[(n - j - k - l - m)/2]}], {j, k, Floor[(n - k - l - m)/3]}], {k, l, Floor[(n - l - m)/4]}], {l, m, Floor[(n - m)/5]}], {m, Floor[n/6]}], {n, 0, 50}]
Formula
a(n) = Sum_{m=1..floor(n/6)} Sum_{l=m..floor((n-m)/5)} Sum_{k=l..floor((n-l-m)/4)} Sum_{j=k..floor((n-k-l-m)/3)} Sum_{i=j..floor((n-j-k-l-m)/2)} c(m) * c(l) * c(k) * c(j) * c(i) * c(n-i-j-k-l-m) * (n-i-j-k-l-m), where c = A010051.