A261985 Sum of the smaller parts of the partitions of n into two squarefree parts.

0, 1, 1, 3, 2, 4, 3, 6, 5, 8, 6, 14, 11, 11, 8, 17, 18, 16, 13, 32, 25, 27, 19, 39, 32, 39, 35, 58, 40, 47, 32, 61, 47, 65, 41, 93, 58, 67, 54, 95, 73, 80, 89, 130, 109, 94, 87, 142, 110, 106, 102, 203, 129, 130, 115, 189, 148, 151, 137, 232, 170, 165, 169

Offset

1,4

Comments

Sum of the widths of the distinct rectangles with squarefree length and width such that L + W = n, W <= L. For example a(16) = 17; the rectangles are 1 X 15, 2 X 14, 3 X 13, 5 X 11, 6 X 10 and the sum of the widths of these rectangles gives 1 + 2 + 3 + 5 + 6 = 17. - Wesley Ivan Hurt, Nov 02 2017

Links

Altug Alkan, 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)} (i * mu(i)^2 * mu(n-i)^2), where mu is the Moebius function (A008683).

a(n) = A262351(n) - A261927(n).

a(n) = A303221(n)/n. - Wesley Ivan Hurt, Jan 05 2024

Example

a(4)=3; there are two partitions of 4 into two squarefree parts: (3,1) and (2,2). The sum of the smaller parts of these partitions is 1+2=3.
a(6)=4; there are two partitions of 6 into two squarefree parts: (5,1) and (3,3). The sum of the smaller parts is 1+3=4.

Maple

with(numtheory): A261985:=n->add(i*mobius(i)^2*mobius(n-i)^2, i=1..floor(n/2)): seq(A261985(n), n=1..100);

Mathematica

Table[Sum[i*MoebiusMu[i]^2*MoebiusMu[n - i]^2, {i, Floor[n/2]}], {n, 100}]

Prog

(PARI) a(n) = sum(i=1, n\2, i*moebius(i)^2*moebius(n-i)^2); \\ Altug Alkan, Jan 01 2018

Crossrefs

Cf. A008683, A071068, A261927, A262351, A303221.

Sequence in context: A335507 A195459 A133131 * A026923 A058708 A141731

Adjacent sequences: A261982 A261983 A261984 * A261986 A261987 A261988

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Oct 02 2015

Status

approved