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A054521
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Triangle T(n,k): T(n,k)=1 if GCD(n,k) = 1, T(n,k)=0 otherwise (n >= 1, 1 <= k <= n).
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52
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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
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OFFSET
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1,1
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COMMENTS
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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
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LINKS
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Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened
Jakub Jaroslaw Ciaston, A054531 vs A164306 (plot shows these ones)
Index entries for characteristic functions
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FORMULA
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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
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EXAMPLE
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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
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MAPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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Cf. A051731, A054522, A215200.
Cf. A050873, A063524.
Sequence in context: A136442 A168030 A128431 * A014240 A014471 A071028
Adjacent sequences: A054518 A054519 A054520 * A054522 A054523 A054524
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KEYWORD
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nonn,tabl
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AUTHOR
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N. J. A. Sloane, Apr 09 2000
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STATUS
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approved
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