A262869 Number of squarefree numbers appearing among the smaller parts of the partitions of n into two parts.

0, 1, 1, 2, 2, 3, 3, 3, 3, 4, 4, 5, 5, 6, 6, 6, 6, 6, 6, 7, 7, 8, 8, 8, 8, 9, 9, 10, 10, 11, 11, 11, 11, 12, 12, 12, 12, 13, 13, 13, 13, 14, 14, 15, 15, 16, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 17, 18, 18, 19, 19, 20, 20, 20, 20, 21, 21, 22, 22, 23, 23

Offset: 1

Author

Wesley Ivan Hurt, Oct 03 2015

Keywords

• nonn
• easy

Comments

Number of distinct rectangles with integer length and squarefree width such that L + W = n, W <= L. For example, a(14) = 6; the rectangles are 13 X 1, 12 X 2, 11 X 3, 9 X 5, 8 X 6, 7 X 7. - Wesley Ivan Hurt, Nov 04 2017

Examples

			a(5)=2; there are two partitions of 5 into two parts: (4,1) and (3,2). Both of the smaller parts are squarefree, thus a(5)=2.
a(6)=3; there are three partitions of 6 into two parts: (5,1), (4,2) and (3,3). Among the three smaller parts, all are squarefree, thus a(6)=3.
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
• Index entries for sequences related to partitions

Crossrefs

Cf. A008683, A013928, A071068, A261985, A262351, A262868, A262870, A262871, A262991, A262992.

Programs

Maple

with(numtheory): A262869:=n->add(mobius(i)^2, i=1..floor(n/2)): seq(A262869(n), n=1..100);

Mathematica

Table[Sum[MoebiusMu[i]^2, {i, Floor[n/2]}], {n, 100}]
Table[Count[IntegerPartitions[n,{2}][[All,2]],?SquareFreeQ],{n,80}] (* _Harvey P. Dale, Oct 17 2021 *)

PARI

a(n) = sum(i=1, n\2, moebius(i)^2); \\ Michel Marcus, Oct 04 2015

PARI

a(n)=my(s); n\=2; forsquarefree(k=1, sqrtint(n), s += n\k[1]^2*moebius(k)); s \\ Charles R Greathouse IV, Jan 08 2018

Formula

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

a(n) = A262991(n) - A262868(n).

a(n) = A013928(floor(n/2)+1). - Georg Fischer, Nov 29 2022