OFFSET
1,4
COMMENTS
Number of distinct rectangles with squarefree length and integer width such that L + W = n, W <= L. For example, a(14) = 4; the rectangles are 1 X 13, 3 X 11, 4 X 10 and 7 X 7. - Wesley Ivan Hurt, Nov 02 2017
a(10) = 3, a(100) = 30, a(10^3) = 302, a(10^4) = 3041, a(10^5) = 30393, a(10^6) = 303968, a(10^7) = 3039658, a(10^8) = 30396350, a(10^9) = 303963598, a(10^10) = 3039635373, a(10^11) = 30396355273, a(10^12) = 303963551068, a(10^13) = 3039635509338, a(10^14) = 30396355094469, a(10^15) = 303963550926043, a(10^16) = 3039635509271763, a(10^17) = 30396355092700721, and a(10^18) = 303963550927014110. The limit of a(n)/n is 3/Pi^2. - Charles R Greathouse IV, Nov 04 2017
LINKS
Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 150 terms from G. C. Greubel)
FORMULA
a(n) = Sum_{i=1..floor(n/2)} mu(n-i)^2, where mu is the Möbius function A008683.
a(n) ~ 3*n/Pi^2. - Charles R Greathouse IV, Nov 04 2017
EXAMPLE
a(4)=2; there are two partitions of 4 into two parts: (3,1) and (2,2). Both of the larger parts are squarefree, thus a(4)=2.
a(5)=1; there are two partitions of 5 into two parts: (4,1) and (3,2). Among the larger parts, only 3 is squarefree, thus a(5)=1.
MATHEMATICA
Table[Sum[MoebiusMu[n - i]^2, {i, Floor[n/2]}], {n, 100}]
Table[Count[IntegerPartitions[n, {2}][[All, 1]], _?SquareFreeQ], {n, 80}] (* Harvey P. Dale, Jan 03 2022 *)
PROG
(PARI) a(n) = sum(i=1, n\2, moebius(n-i)^2); \\ Michel Marcus, Oct 04 2015
(PARI) f(n)=my(s); forfactored(k=1, sqrtint(n), s+=n\k[1]^2*moebius(k)); s
a(n)=n--; f(n) - f(n\2) \\ Charles R Greathouse IV, Nov 04 2017
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Wesley Ivan Hurt, Oct 03 2015
STATUS
approved