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A368058
Sum of the smaller parts of the partitions of n into two distinct parts with larger part prime.
2
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
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
KEYWORD
nonn
AUTHOR
Wesley Ivan Hurt, Dec 09 2023
STATUS
approved