A262868 Number of squarefree numbers appearing among the larger parts of the partitions of n into two parts.

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, 10, 11, 11, 12, 12, 13, 13, 14, 14, 15, 14, 14, 14, 15, 15, 15, 15, 16, 15, 16, 16, 17, 17, 18, 18, 19, 18, 19, 19, 19, 19, 20, 20, 21, 20, 21, 21, 22, 22

Offset: 1

Author

Wesley Ivan Hurt, Oct 03 2015

Keywords

• nonn
• easy

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

Examples

			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.
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 150 terms from G. C. Greubel)
• Index entries for sequences related to partitions

Crossrefs

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

Programs

Maple

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

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 *)

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

Formula

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

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

a(n) ~ 3*n/Pi^2. - Charles R Greathouse IV, Nov 04 2017