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A295284 Number of partitions of n into two distinct parts such that the larger part is nonsquarefree. 1
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 (list; graph; refs; listen; history; text; internal format)
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).

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}]

PROG

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

CROSSREFS

Cf. 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

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Last modified May 24 18:34 EDT 2019. Contains 323534 sequences. (Running on oeis4.)