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

Author

Wesley Ivan Hurt, Dec 09 2023

Keywords

• nonn

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Index entries for sequences related to partitions

Crossrefs

Cf. A010051, A294111, A294487, A294602.

Programs

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}]

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