A294487 Sum of the lengths of the distinct rectangles with prime length and integer width such that L + W = n, W < L.

0, 0, 2, 3, 3, 5, 5, 12, 12, 7, 7, 18, 18, 24, 24, 24, 24, 41, 41, 60, 60, 49, 49, 72, 72, 59, 59, 59, 59, 88, 88, 119, 119, 102, 102, 102, 102, 120, 120, 120, 120, 161, 161, 204, 204, 181, 181, 228, 228, 228, 228, 228, 228, 281, 281, 281, 281, 252, 252, 311

Offset: 1

Author

Wesley Ivan Hurt, Oct 31 2017

Keywords

• nonn
• easy

Comments

Sum of the largest parts of the partitions of n into two distinct parts with largest part prime.

Examples

			a(14) = 24; the rectangles are 1 X 13 and 3 X 11 (7 X 7 is not considered since W < L). The sum of the lengths is then 13 + 11 = 24.
		

Links

• Index entries for sequences related to partitions

Crossrefs

Cf. A010051, A243485, A294602, A368058.

Programs

Mathematica

Table[ Sum[(n - i)*(PrimePi[n - i] - PrimePi[n - i - 1]), {i, Floor[(n-1)/2]}], {n, 60}]

PARI

a(n) = sum(i=1, (n-1)\2, (n-i)*isprime(n-i)); \\ Michel Marcus, Nov 08 2017

Formula

a(n) = Sum_{i=1..floor((n-1)/2)} (n-i) * A010051(n-i).

a(n) = n*A294602(n) - A368058(n). - Wesley Ivan Hurt, Dec 09 2023