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Moebius function applied to divisors of n, table read by rows.
9

%I #9 Dec 10 2014 19:31:13

%S 1,1,-1,1,-1,1,-1,0,1,-1,1,-1,-1,1,1,-1,1,-1,0,0,1,-1,0,1,-1,-1,1,1,

%T -1,1,-1,-1,0,1,0,1,-1,1,-1,-1,1,1,-1,-1,1,1,-1,0,0,0,1,-1,1,-1,-1,1,

%U 0,0,1,-1,1,-1,0,-1,1,0,1,-1,-1,1,1,-1,-1,1,1,-1

%N Moebius function applied to divisors of n, table read by rows.

%C T(n,k) = A008683(A027750(n,k)), k = 1..A000005(n);

%C T(n,1) = 1; for n > 1: T(n,2) = -1;

%C T(n,A000005(n)) = A008683(n);

%C A048105(n) = number of zeros in row n;

%C A034444(n) = number of nonzero terms in row n;

%C A007875(n) = number of ones in row n.

%H Reinhard Zumkeller, <a href="/A225817/b225817.txt">Rows n = 1..1000 of table, flattened</a>

%e . n | Initial rows | A027750(n,[1..A000005(n)])

%e . -----+-----------------------------------+-- divisors of n: -----------

%e . 1 | 1 | 1

%e . 2 | 1 -1 | 1,2

%e . 3 | 1 -1 | 1,3

%e . 4 | 1 -1 0 | 1,2,4

%e . 5 | 1 -1 | 1,5

%e . 6 | 1 -1 -1 1 | 1,2,3,6

%e . 7 | 1 -1 | 1,7

%e . 8 | 1 -1 0 0 | 1,2,4,8

%e . 9 | 1 -1 0 | 1,3,9

%e . 10 | 1 -1 -1 1 | 1,2,5,10

%e . 11 | 1 -1 | 1,11

%e . 12 | 1 -1 -1 0 1 0 | 1,2,3,4,6,12

%e . 13 | 1 -1 | 1,13

%e . 14 | 1 -1 -1 1 | 1,2,7,14

%e . 15 | 1 -1 -1 1 | 1,3,5,15

%e . 16 | 1 -1 0 0 0 | 1,2,4,8,16

%e . 17 | 1 -1 | 1,17

%e . 18 | 1 -1 -1 1 0 0 | 1,2,3,6,9,18

%e . 19 | 1 -1 | 1,19

%e . 20 | 1 -1 0 -1 1 0 | 1,2,4,5,10,20

%e . 21 | 1 -1 -1 1 | 1,3,7,21

%e . 22 | 1 -1 -1 1 | 1,2,11,22

%e . 23 | 1 -1 | 1,23

%e . 24 | 1 -1 -1 0 1 0 0 0 | 1,2,3,4,6,8,12,24

%e . 25 | 1 -1 0 | 1,5,25

%e . 26 | 1 -1 -1 1 | 1,2,13,26

%e . 27 | 1 -1 0 0 | 1,3,9,27

%e . 28 | 1 -1 0 -1 1 0 | 1,2,4,7,14,28

%e . 29 | 1 -1 | 1,29

%e . 30 | 1 -1 -1 -1 1 1 1 -1 | 1,2,3,5,6,10,15,30 .

%t Table[Map[MoebiusMu, Divisors[n]], {n, 1, 20}] // Grid (* _Geoffrey Critzer_, Dec 10 2014 *)

%o (Haskell)

%o a225817 n k = a225817_tabf !! (n-1) !! (k-1)

%o a225817_row n = a225817_tabf !! (n-1)

%o a225817_tabf = map (map a008683) a027750_tabf

%Y Cf. A000005 (row lengths), A063524 (row sums), A069158 (row products).

%K sign,tabf

%O 1

%A _Reinhard Zumkeller_, Jul 30 2013