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 A262869 Number of squarefree numbers appearing among the smaller parts of the partitions of n into two parts. 6
 0, 1, 1, 2, 2, 3, 3, 3, 3, 4, 4, 5, 5, 6, 6, 6, 6, 6, 6, 7, 7, 8, 8, 8, 8, 9, 9, 10, 10, 11, 11, 11, 11, 12, 12, 12, 12, 13, 13, 13, 13, 14, 14, 15, 15, 16, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 17, 18, 18, 19, 19, 20, 20, 20, 20, 21, 21, 22, 22, 23, 23 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS Number of distinct rectangles with integer length and squarefree width such that L + W = n, W <= L. For example, a(14) = 6; the rectangles are 13 X 1, 12 X 2, 11 X 3, 9 X 5, 8 X 6, 7 X 7. - Wesley Ivan Hurt, Nov 04 2017 LINKS Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 FORMULA a(n) = Sum_{i=1..floor(n/2)} mu(i)^2, where mu is the Möebius function (A008683). a(n) = A262991(n) - A262868(n). EXAMPLE a(5)=2; there are two partitions of 5 into two parts: (4,1) and (3,2). Both of the smaller parts are squarefree, thus a(5)=2. a(6)=3; there are three partitions of 6 into two parts: (5,1), (4,2) and (3,3). Among the three smaller parts, all are squarefree, thus a(6)=3. MAPLE with(numtheory): A262869:=n->add(mobius(i)^2, i=1..floor(n/2)): seq(A262869(n), n=1..100); MATHEMATICA Table[Sum[MoebiusMu[i]^2, {i, Floor[n/2]}], {n, 100}] Table[Count[IntegerPartitions[n, {2}][[All, 2]], _?SquareFreeQ], {n, 80}] (* Harvey P. Dale, Oct 17 2021 *) PROG (PARI) a(n) = sum(i=1, n\2, moebius(i)^2); \\ Michel Marcus, Oct 04 2015 (PARI) a(n)=my(s); n\=2; forsquarefree(k=1, sqrtint(n), s += n\k^2*moebius(k)); s \\ Charles R Greathouse IV, Jan 08 2018 CROSSREFS Cf. A008683, A071068, A261985, A262351, A262868, A262870, A262871, A262991, A262992. Sequence in context: A225633 A060960 A073642 * A342678 A108356 A239499 Adjacent sequences:  A262866 A262867 A262868 * A262870 A262871 A262872 KEYWORD nonn,easy AUTHOR Wesley Ivan Hurt, Oct 03 2015 STATUS approved

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Last modified July 4 11:31 EDT 2022. Contains 355075 sequences. (Running on oeis4.)