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

1,4

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

Links

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Index entries for sequences related to partitions

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

Example

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.

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A060960 A073642 A356005 * A342678 A108356 A239499

Adjacent sequences: A262866 A262867 A262868 * A262870 A262871 A262872

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Oct 03 2015

Status

approved