A295284 Number of partitions of n into two distinct parts such that the larger part is nonsquarefree.

0, 0, 0, 0, 1, 1, 1, 0, 1, 2, 2, 2, 3, 3, 3, 2, 3, 2, 3, 3, 4, 4, 4, 3, 4, 5, 5, 6, 7, 7, 7, 6, 7, 7, 7, 6, 7, 7, 7, 6, 7, 7, 7, 7, 8, 9, 9, 8, 9, 9, 10, 10, 11, 10, 11, 10, 11, 11, 11, 11, 12, 12, 12, 12, 13, 13, 13, 13, 14, 14, 14, 13, 14, 14, 14, 15, 16

Offset

1,10

Comments

Number of distinct rectangles with nonsquarefree length and integer width such that L + W = n, W < L. For example, a(8) = 0 since none of the rectangles 1 X 7, 2 X 6 and 3 X 5 (4 X 4 is not considered since we have W < L) have nonsquarefree length, i.e., 7,6,5 are all squarefree. a(10) = 2 since there are two rectangles with nonsquarefree length, namely 1 X 9 and 2 X 8.

Links

Table of n, a(n) for n=1..77.

Index entries for sequences related to partitions

Formula

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

Example

a(14) = 3; the partitions of 14 into two parts are (13,1), (12,2), (11,3), (10,4), (9,5), (8,6), (7,7). Three of the larger parts of these partitions are nonsquarefree: 12, 9 and 8, so a(14) = 3.

Maple

A295284:=n->add(1-numtheory[mobius](n-i)^2, i=1..floor((n-1)/2)): seq(A295284(n), n=1..100);

Mathematica

Table[Sum[1 - MoebiusMu[n - k]^2, {k, Floor[(n - 1)/2]}], {n, 80}]
Table[Count[IntegerPartitions[n, {2}], _?(#[[1]]!=#[[2]]&&!SquareFreeQ[ #[[1]]]&)], {n, 80}] (* Harvey P. Dale, Feb 02 2021 *)

Prog

(PARI) a(n) = sum(i=1, floor((n-1)/2), 1 - moebius(n-i)^2) \\ Iain Fox, Dec 08 2017

Crossrefs

Cf. A008683, A008966, A294235.

Sequence in context: A064740 A080967 A078570 * A292995 A037179 A127971

Adjacent sequences: A295281 A295282 A295283 * A295285 A295286 A295287

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Nov 19 2017

Status

approved