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

1,4

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)

Index entries for sequences related to partitions

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

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.

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 *)

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

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

Sequence in context: A333708 A132203 A158925 * A342097 A259357 A031265

Adjacent sequences: A262865 A262866 A262867 * A262869 A262870 A262871

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Oct 03 2015

Status

approved