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A308925
Sum of the largest parts in the partitions of n into 6 primes.
7
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
0,13
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.
a(n) = A308919(n) - A308920(n) - A308921(n) - A308922(n) - A308923(n) - A308924(n).
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}]
KEYWORD
nonn
AUTHOR
Wesley Ivan Hurt, Jun 30 2019
STATUS
approved