A355197 Sum of the middle parts of the partitions of n into exactly 3 prime parts.

0, 0, 0, 0, 0, 0, 2, 2, 3, 5, 3, 5, 8, 8, 5, 12, 10, 17, 8, 15, 12, 26, 10, 24, 19, 33, 23, 48, 23, 57, 16, 54, 31, 80, 16, 71, 36, 93, 42, 112, 26, 122, 31, 120, 48, 145, 23, 148, 48, 160, 59, 190, 42, 218, 39, 196, 71, 240, 35, 246, 62, 252, 108, 310, 53, 366, 59, 340, 96

Offset

0,7

Links

Table of n, a(n) for n=0..68.

Index entries for sequences related to partitions

Formula

a(n) = Sum_{j=1..floor(n/3)} Sum_{i=j..floor((n-j)/2)} c(i) * c(j) * c(n-i-j) * i, where c = A010051.

a(n) = A355199(n) - A355196(n) - A355198(n).

Example

a(9) = 5; since 9 can be written as the sum of 3 primes in two ways: 2+2+5 = 3+3+3 and the sum of the middle parts of these partitions is 2+3 = 5, we have a(9) = 5.

Mathematica

Table[Sum[Sum[i (PrimePi[i] - PrimePi[i - 1]) (PrimePi[j] - PrimePi[j - 1])(PrimePi[n - i - j] - PrimePi[n - i - j - 1]), {i, j, Floor[(n - j)/2]}], {j, Floor[n/3]}], {n, 0, 80}]

Crossrefs

Cf. A010051, A068307, A355196, A355198, A355199.

Sequence in context: A317266 A067330 A202874 * A199512 A303969 A304931

Adjacent sequences: A355194 A355195 A355196 * A355198 A355199 A355200

Keyword

nonn

Author

Wesley Ivan Hurt, Jun 23 2022

Status

approved