A294247 Sum of the parts in the partitions of n into exactly two distinct squarefree parts.

0, 0, 3, 4, 5, 6, 14, 24, 18, 10, 22, 36, 39, 28, 45, 80, 68, 72, 57, 100, 84, 88, 92, 168, 125, 104, 135, 168, 145, 120, 155, 256, 198, 204, 210, 396, 259, 228, 273, 440, 328, 294, 387, 528, 450, 322, 376, 624, 490, 400, 357, 676, 530, 540, 385, 728, 570

Offset: 1

Author

Wesley Ivan Hurt, Oct 25 2017; recomputed Oct 26 2017 with thanks to Andrey Zabolotskiy

Keywords

• nonn
• easy

Comments

Sum of the semiperimeters of the distinct rectangles with squarefree length and width such that L + W = n, W < L.

Examples

			For n = 4,5,6,7 the partitions are respectively 1+3 (sum a(4) = 4), 2+3 (sum 5), 1+5 (sum 6), 1+6 and 2+5 (sum 7+7=14). - _N. J. A. Sloane_, Oct 28 2017
		

Links

• Index entries for sequences related to partitions

Crossrefs

Cf. A008683, A262351.

Programs

Mathematica

Table[n*Sum[MoebiusMu[i]^2*MoebiusMu[n - i]^2, {i, Floor[(n-1)/2]}], {n, 80}]

Python

from sympy import mobius
def a(n): return n*sum(mobius(i)**2*mobius(n - i)**2 for i in range(1, ((n - 1)//2) + 1))
print([a(n) for n in range(1, 51)]) # Indranil Ghosh, Nov 07 2017

R

require(numbers)
a <- function(n) {
  if (n<3) return(0)
  S <- numeric()
  for (i in 1:floor((n-1)/2)) S <- c(S, moebius(i)^2*moebius(n-i)^2)
  return(n*sum(S))
}
sapply(1:100, a) # Indranil Ghosh, Nov 07 2017

Formula

a(n) = n * Sum_{i=1..floor((n-1)/2)} mu(i)^2 * mu(n-i)^2, where mu(n) is the Möbius function (A008683).