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A051731 Triangle read by rows: T(n,k) = 1 if k divides n, T(n,k) = 0 otherwise. 249
1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Row sums are A000005. Diagonal sums are A032741(n+2). Might be called a Mobius matrix. Binomial transform (product by binomial matrix) is A101508. - Paul Barry, Dec 05 2004

A054525 is the inverse of this triangle (as lower triangular matrix). - Gary W. Adamson, Apr 15 2007

A049820(n) = number of zeros in n-th row. - Reinhard Zumkeller, Mar 09 2010

The determinant of this matrix where T(n,n) has been swapped with T(1,k) is equal to the n-th term of the mobius function. - Mats Granvik, Jul 21 2012

The k-th column of this table is the k-dimensional matrix formulation of the inverse discrete Fourier transform of the all ones sequence 1,1,1,... Example; the 6th column is equal to: (1/6)*Sum_{k=1..6} cos(2*Pi*(n-1)*(k-1)/6) (shifted one step in the n direction). - Mats Granvik, Nov 11 2012

LINKS

Charles R Greathouse IV, Rows n = 1..100, flattened

Mats Granvik, Illustration

Mats Granvik, Better illustration

Jeffrey Ventrella, Divisor Plot

FORMULA

{T(n,k)*k, k=1..n} setminus {0} = divisors of n; Sum_{k=1..n} T(n,k)*k^i = sigma[i](n) = sum of the i-th power of positive divisors of n; Sum_{k=1..n} T(n,k) = A000005(n), Sum_{k=1..n} T(n,k)*k = A000203(n).

T(n,k) = T(n-k, k) for k <= n/2, T(n,k) = 0 for n/2 < k <= n-1, T(n,n) = 1.

Rows given by A074854 converted to binary. Example: A074854(4) = 13 = 1101_2; row 4 = 1, 1, 0, 1. - Philippe Deléham, Oct 04 2003

Columns have g.f.: x^k/(1-x^(k+1)) (k >= 0). - Paul Barry, Dec 05 2004

Matrix inverse of triangle A054525, where A054525(n, k) = MoebiusMu(n/k) if k|n, 0 otherwise. - Paul D. Hanna, Jan 09 2006

Equals A129372 * A115361 as infinite lower triangular matrices. - Gary W. Adamson, Apr 15 2007

From Gary W. Adamson, May 10 2007: (Start)

This triangle * [1, 2, 3, ...] = sigma(n), A000203: (1, 3, 4, 7, 6, 12, 8, ...).

This triangle * [1/1, 1/2, 1/3, ...] = sigma(n)/n: (1/1, 3/2, 4/3, 7/4, 6/5, ...). (End)

T(n,k) = 0^(n mod k). - Reinhard Zumkeller, Nov 01 2009

T(n,k) = A000007(A048158(n,k)). - Reinhard Zumkeller, Nov 01 2009

T(n,k) = A172119(n) mod 2. - Mats Granvik, Jan 26 2010

T(n,k) = A175105(n) mod 2. - Mats Granvik, Feb 10 2010

Jeffrey O. Shallit kindly provided a clarification along with a proof of this formula T(n,1) = 1, k > 1: T(n,k) = Sum_{i=1..k-1} (T(n-i,k-1) - T(n-i,k)). - Mats Granvik, Feb 16 2010

T(n,k) = (A181116/A181117)*(A181116/A181117). - Mats Granvik, Oct 04 2010

T(n,k) = lim_{r -> infinity} ((cos(2*Pi*n/k) + 1)/2)^r.

T(n,k) = lim_{r -> infinity} (1-(sin(Pi*n/k)/(Pi*n/k))^2)^r. Setting r = 1 and k = 1 the expression is the same as the term in the Montgomery pair correlation conjecture. - Mats Granvik, Dec 09 2011

EXAMPLE

The triangle T(n,k) begins:

n\k 1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 ...

1:  1

2:  1  1

3:  1  0  1

4:  1  1  0  1

5:  1  0  0  0  1

6:  1  1  1  0  0  1

7:  1  0  0  0  0  0  1

8:  1  1  0  1  0  0  0  1

9:  1  0  1  0  0  0  0  0  1

10: 1  1  0  0  1  0  0  0  0  1

11: 1  0  0  0  0  0  0  0  0  0  1

12: 1  1  1  1  0  1  0  0  0  0  0  1

13: 1  0  0  0  0  0  0  0  0  0  0  0  1

14: 1  1  0  0  0  0  1  0  0  0  0  0  0  1

15: 1  0  1  0  1  0  0  0  0  0  0  0  0  0  1

... Reformatted and extended. - Wolfdieter Lang, Nov 12 2014

MAPLE

A051731 := proc(n, k)

        if n mod k =0 then

                1;

        else

                0;

        end if;

end proc: # R. J. Mathar, Jul 14 2012

MATHEMATICA

Flatten[Table[If[Mod[n, k] == 0, 1, 0], {n, 20}, {k, n}]]

PROG

(PARI) for(n=1, 9, for(k=1, n, print1(!(n%k)", "))) \\ Charles R Greathouse IV, Mar 14, 2012

(Haskell)

a051731 n k = 0 ^ mod n k

a051731_row n = a051731_tabl !! (n-1)

a051731_tabl = map (map a000007) a048158_tabl

-- Reinhard Zumkeller, Aug 13 2013

CROSSREFS

Cf. A000005, A000203, A074854, A054525, A129372, A115361.

A077049 and A077051 are other versions of this matrix.

Cf. A002260.

Cf. A134546 (A004736 * T, matrix multiplication).

Partial sums per row: A243987.

Sequence in context: A110471 A174854 A103994 * A135839 A071022 A155076

Adjacent sequences:  A051728 A051729 A051730 * A051732 A051733 A051734

KEYWORD

easy,nice,nonn,tabl,changed

AUTHOR

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de)

STATUS

approved

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Last modified August 21 13:36 EDT 2017. Contains 290890 sequences.