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

1,3

Comments

One-half of the sum of the perimeters of the distinct rectangles with squarefree length and width such that L + W = n, W < L.

Links

Table of n, a(n) for n=1..57.

Index entries for sequences related to partitions

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).

Example

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

Mathematica

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

Prog

(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

Crossrefs

Cf. A008683, A262351.

Sequence in context: A107228 A355706 A351711 * A346310 A083401 A281829

Adjacent sequences: A294244 A294245 A294246 * A294248 A294249 A294250

Keyword

nonn,easy

Author

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

Status

approved