cp's OEIS Frontend

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

%I A294247 #29 May 03 2024 13:17:57
%S A294247 0,0,3,4,5,6,14,24,18,10,22,36,39,28,45,80,68,72,57,100,84,88,92,168,
%T A294247 125,104,135,168,145,120,155,256,198,204,210,396,259,228,273,440,328,
%U A294247 294,387,528,450,322,376,624,490,400,357,676,530,540,385,728,570
%N A294247 Sum of the parts in the partitions of n into exactly two distinct squarefree parts.
%C A294247 Sum of the semiperimeters of the distinct rectangles with squarefree length and width such that L + W = n, W < L.
%H A294247 <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F A294247 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).
%e A294247 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
%t A294247 Table[n*Sum[MoebiusMu[i]^2*MoebiusMu[n - i]^2, {i, Floor[(n-1)/2]}], {n, 80}]
%o A294247 (Python)
%o A294247 from sympy import mobius
%o A294247 def a(n): return n*sum(mobius(i)**2*mobius(n - i)**2 for i in range(1, ((n - 1)//2) + 1))
%o A294247 print([a(n) for n in range(1, 51)]) # _Indranil Ghosh_, Nov 07 2017
%o A294247 (R)
%o A294247 require(numbers)
%o A294247 a <- function(n) {
%o A294247   if (n<3) return(0)
%o A294247   S <- numeric()
%o A294247   for (i in 1:floor((n-1)/2)) S <- c(S, moebius(i)^2*moebius(n-i)^2)
%o A294247   return(n*sum(S))
%o A294247 }
%o A294247 sapply(1:100, a) # _Indranil Ghosh_, Nov 07 2017
%Y A294247 Cf. A008683, A262351.
%K A294247 nonn,easy
%O A294247 1,3
%A A294247 _Wesley Ivan Hurt_, Oct 25 2017; recomputed Oct 26 2017 with thanks to _Andrey Zabolotskiy_