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%I #36 Oct 17 2019 14:26:10
%S 1,2,1,3,3,1,4,2,4,1,5,5,5,5,1,6,3,2,3,6,1,7,7,7,7,7,7,1,8,4,8,2,8,4,
%T 8,1,9,9,3,9,9,3,9,9,1,10,5,10,5,2,5,10,5,10,1,11,11,11,11,11,11,11,
%U 11,11,11,1,12,6,4,3,12,2,12,3,4,6,12,1,13,13,13,13,13
%N Triangular array T read by rows: T(n,k) = n/gcd(n,k) (n >= 1, 1 <= k <= n).
%C Sum of n-th row = A057660(n). - _Reinhard Zumkeller_, Aug 12 2009
%C Read as a linear sequence, this is conjectured to be the length of the shortest cycle of pebble-moves among the partitions of n (cf. A201144). - _Andrew V. Sutherland_, Nov 27 2011
%C The triangle of fractions A226314(i,j)/A054531(i,j) is an efficient way to enumerate the rationals [Fortnow]. - _N. J. A. Sloane_, Jun 09 2013
%H Reinhard Zumkeller, <a href="/A054531/b054531.txt">Rows n = 1..120 of triangle, flattened</a>
%H Lance Fortnow, <a href="http://blog.computationalcomplexity.org/2004/03/counting-rationals-quickly.html">Counting the Rationals Quickly</a>, Computational Complexity Weblog, Monday, March 01, 2004.
%H R. J. Mathar, <a href="http://www.vixra.org/abs/1406.0183">Plots of cycle graphs of the finite groups up to order 36</a>, (2015).
%H Yoram Sagher, <a href="http://www.jstor.org/stable/2324846">Counting the rationals</a>, Amer. Math. Monthly, 96 (1989), p. 823. Math. Rev. 90i:04001.
%e Triangle begins
%e 1;
%e 2, 1;
%e 3, 3, 1;
%e 4, 2, 4, 1;
%e 5, 5, 5, 5, 1;
%e 6, 3, 2, 3, 6, 1;
%e 7, 7, 7, 7, 7, 7, 1;
%e 8, 4, 8, 2, 8, 4, 8, 1;
%e 9, 9, 3, 9, 9, 3, 9, 9, 1;
%e 10, 5, 10, 5, 2, 5, 10, 5, 10, 1;
%e 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 1;
%e 12, 6, 4, 3, 12, 2, 12, 3, 4, 6, 12, 1;
%e 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 1;
%t Table[n/GCD[n,k], {n,1,10}, {k,1,n}]//Flatten (* _G. C. Greubel_, Sep 13 2017 *)
%o (Haskell)
%o a054531 n k = div n $ gcd n k
%o a054531_row n = a054531_tabl !! (n-1)
%o a054531_tabl = zipWith (\u vs -> map (div u) vs) [1..] a050873_tabl
%o -- _Reinhard Zumkeller_, Jun 10 2013
%o (PARI) for(n=1,10, for(k=1,n, print1(n/gcd(n,k), ", "))) \\ _G. C. Greubel_, Sep 13 2017
%Y Cf. A050873, A164306, A226314, A277227 (row reversed, k=0..n-1).
%K nonn,tabl,frac,easy
%O 1,2
%A _N. J. A. Sloane_, Apr 09 2000
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