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

Author

Wesley Ivan Hurt, Jun 30 2019

Keywords

• nonn

Links

• Robert Israel, Table of n, a(n) for n = 0..2000
• Index entries for sequences related to partitions

Crossrefs

Cf. A010051, A259196, A308919, A308920, A308921, A308922, A308923, A308924.

Programs

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.

a(n) = A308919(n) - A308920(n) - A308921(n) - A308922(n) - A308923(n) - A308924(n).