A054521 Triangle read by rows: T(n,k) = 1 if gcd(n, k) = 1, T(n,k) = 0 otherwise (n >= 1, 1 <= k <= n).

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

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 T(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

This triangle is the j = 1 sub-triangle of A349221(n,k) = Sum_{j>=1} [k|binomial(n-1,k-1) AND gcd(n,k) = j], n >= 1, 1 <= k <= n, where [] is the Iverson bracket. - Richard L. Ollerton, Dec 14 2021

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*T(2,1) = 2 = T(3,1) + T(3,2) = phi(3). n = 4: 2*(T(3,1) + T(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.

Cf. A349221.

Sequence in context: A128431 A290452 A215200 * A349221 A338354 A014240

Adjacent sequences: A054518 A054519 A054520 * A054522 A054523 A054524

Keyword

nonn,tabl

Author

N. J. A. Sloane, Apr 09 2000

Status

approved