A262868 - OEIS

%I #47 Jan 03 2022 14:51:17

%S 0,1,1,2,1,2,2,3,3,3,3,4,3,4,4,5,5,6,6,7,6,7,7,8,8,8,8,8,7,8,8,9,9,10,

%T 10,11,11,12,12,13,13,14,14,15,14,14,14,15,15,15,15,16,15,16,16,17,17,

%U 18,18,19,18,19,19,19,19,20,20,21,20,21,21,22,22

%N Number of squarefree numbers appearing among the larger parts of the partitions of n into two parts.

%C 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

%C 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

%H Charles R Greathouse IV, <a href="/A262868/b262868.txt">Table of n, a(n) for n = 1..10000</a> (first 150 terms from G. C. Greubel)

%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>

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

%F a(n) = A262991(n) - A262869(n).

%F a(n) ~ 3*n/Pi^2. - _Charles R Greathouse IV_, Nov 04 2017

%e 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.

%e 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.

%p with(numtheory): A262868:=n->add(mobius(n-i)^2, i=1..floor(n/2)): seq(A262868(n), n=1..100);

%t Table[Sum[MoebiusMu[n - i]^2, {i, Floor[n/2]}], {n, 100}]

%t Table[Count[IntegerPartitions[n,{2}][[All,1]],_?SquareFreeQ],{n,80}] (* _Harvey P. Dale_, Jan 03 2022 *)

%o (PARI) a(n) = sum(i=1, n\2, moebius(n-i)^2); \\ _Michel Marcus_, Oct 04 2015

%o (PARI) f(n)=my(s); forfactored(k=1,sqrtint(n),s+=n\k[1]^2*moebius(k)); s

%o a(n)=n--; f(n) - f(n\2) \\ _Charles R Greathouse IV_, Nov 04 2017

%Y Cf. A008683, A071068, A261985, A262351, A262869, A262870, A262871, A262991, A262992, A013928.

%K nonn,easy

%O 1,4

%A _Wesley Ivan Hurt_, Oct 03 2015