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A294247 Sum of the parts in the partitions of n into exactly two distinct squarefree parts. 2
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 (list; graph; refs; listen; history; text; internal format)
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, floor

def a(n): return n*sum([mobius(i)**2*mobius(n - i)**2 for i in range(1, floor((n - 1)//2) + 1)])

print(map(a, range(1, 101))) # 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: A299496 A070981 A107228 * A083401 A281829 A083400

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

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Last modified November 15 06:11 EST 2019. Contains 329144 sequences. (Running on oeis4.)