

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
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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((n1)/2)} (1  mu(ni)^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(1numtheory[mobius](ni)^2, i=1..floor((n1)/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((n1)/2), 1  moebius(ni)^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



