A368058 Sum of the smaller parts of the partitions of n into two distinct parts with larger part prime.

0, 0, 1, 1, 2, 1, 2, 4, 6, 3, 4, 6, 8, 4, 6, 8, 10, 13, 16, 20, 24, 17, 20, 24, 28, 19, 22, 25, 28, 32, 36, 41, 46, 34, 38, 42, 46, 32, 36, 40, 44, 49, 54, 60, 66, 49, 54, 60, 66, 72, 78, 84, 90, 97, 104, 111, 118, 96, 102, 109, 116, 93, 100, 107, 114, 121, 128, 136, 144, 152, 160

Offset

1,5

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Index entries for sequences related to partitions

Formula

a(n) = Sum_{i=1..floor((n-1)/2)} i * c(n - i), where c is the prime characteristic (A010051).

a(n) = n*A294602(n) - A294487(n). - Wesley Ivan Hurt, Dec 09 2023

Maple

N:= 100: # for a(1) .. a(N)
V:= Vector(N):
for i from 1 do
  p:= ithprime(i);
  if p >= N then break fi;
  m:= min(2*p-1, N);
  V[p+1..m]:= V[p+1..m] + <$1..m-p>
od:
convert(V, list); # Robert Israel, Jan 26 2024

Mathematica

Table[Sum[i (PrimePi[n - i] - PrimePi[n - i - 1]), {i, Floor[(n - 1)/2]}], {n, 100}]

Crossrefs

Cf. A010051, A294111, A294487, A294602.

Sequence in context: A201635 A179787 A358918 * A324225 A214739 A296159

Adjacent sequences: A368055 A368056 A368057 * A368059 A368060 A368061

Keyword

nonn

Author

Wesley Ivan Hurt, Dec 09 2023

Status

approved