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A261985
Sum of the smaller parts of the partitions of n into two squarefree parts.
9
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
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
KEYWORD
nonn,easy
AUTHOR
Wesley Ivan Hurt, Oct 02 2015
STATUS
approved