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A054521 Triangle T(n,k): T(n,k)=1 if GCD(n,k) = 1, T(n,k)=0 otherwise (n >= 1, 1 <= k <= n). 51
1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Row sums = phi(n), A000010: (1, 1, 2, 2, 4, 2, 6, ...). - Gary W. Adamson, May 20 2007

Characteristic function of A169581: a(A169581(n))=1; a(A169582(n))=0. - Reinhard Zumkeller, Dec 02 2009

The function T(n,k) = T(k,n) is defined for k>n but only the values for 1<=k<=n as a triangular array are listed here.

T(n,k) = |K(n-k|k)| where K(i|j) is the Kronecker symbol. - Peter Luschny, Aug 05 2012

Twice the sum over the antidiagonals, starting with entry a(n-1,1), for n >= 3, is the same as the row n sum (i.e., phi(n): 2*Sum_{k=1..floor(n/2)} T(n-k,k) = phi(n), n >= 3). - Wolfdieter Lang, Apr 26 2013

The number of zeros in the n-th row of the triangle is cototient(n) = A051953(n). - Omar E. Pol, Apr 21 2017

LINKS

Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened

Jakub Jaroslaw Ciaston, A054531 vs A164306 (plot shows these ones)

Index entries for characteristic functions

FORMULA

T(n,k) = A063524(A050873(n,k)). - Reinhard Zumkeller, Dec 02 2009, corrected Sep 03 2015

T(n,k) = A054431(n,k) = A054431(k,n). - R. J. Mathar, Jul 21 2016

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  0

3:   1  1  0

4:   1  0  1  0

5:   1  1  1  1  0

6:   1  0  0  0  1  0

7:   1  1  1  1  1  1  0

8:   1  0  1  0  1  0  1  0

9:   1  1  0  1  1  0  1  1  0

10:  1  0  1  0  0  0  1  0  1  0

11:  1  1  1  1  1  1  1  1  1  1  0

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

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

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

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

... (Reformatted by Wolfdieter Lang, Apr 26 2013)

Sums over antidiagonals: n=3: 2*a(2,1) = 2 = a(3,1) + a(3,2) = phi(3). n=4: 2*(a(3,1) + a(2,2)) = 2 = phi(4), etc. - Wolfdieter Lang, Apr 26 2013

MAPLE

A054521_row := n -> seq(abs(numtheory[jacobi](n-k, k)), k=1..n);

for n from 1 to 13 do A054521_row(n) od; # Peter Luschny, Aug 05 2012

MATHEMATICA

T[ n_, k_] := Boole[ n>0 && k>0 && GCD[ n, k] == 1] (* Michael Somos, Jul 17 2011 *)

T[ n_, k_] := If[ n<1 || k<1, 0, If[ k>n, T[ k, n], If[ k==1, 1, If[ n>k, T[ k, Mod[ n, k, 1]], 0]]] (* Michael Somos, Jul 17 2011 *)

PROG

(PARI) {T(n, k) = n>0 && k>0 && gcd(n, k)==1} /* Michael Somos, Jul 17 2011 */

(Sage)

def A054521_row(n): return [abs(kronecker_symbol(n-k, k)) for k in (1..n)]

for n in (1..13): print A054521_row(n) # Peter Luschny, Aug 05 2012

(Haskell)

a054521 n k = a054521_tabl !! (n-1) !! (k-1)

a054521_row n = a054521_tabl !! (n-1)

a054521_tabl = map (map a063524) a050873_tabl

a054521_list = concat a054521_tabl

-- Reinhard Zumkeller, Sep 03 2015

CROSSREFS

Cf. A051731, A054522, A215200.

Cf. A050873, A063524.

Sequence in context: A136442 A168030 A128431 * A014240 A014471 A071028

Adjacent sequences:  A054518 A054519 A054520 * A054522 A054523 A054524

KEYWORD

nonn,tabl

AUTHOR

N. J. A. Sloane, Apr 09 2000

STATUS

approved

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Last modified February 17 15:58 EST 2018. Contains 299296 sequences. (Running on oeis4.)