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A054531 Triangular array T read by rows: T(n,k) = n/gcd(n,k) (n >= 1, 1 <= k <= n).

%I A054531 #36 Oct 17 2019 14:26:10
%S A054531 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 A054531 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 A054531 11,11,11,1,12,6,4,3,12,2,12,3,4,6,12,1,13,13,13,13,13
%N A054531 Triangular array T read by rows: T(n,k) = n/gcd(n,k) (n >= 1, 1 <= k <= n).
%C A054531 Sum of n-th row = A057660(n). - _Reinhard Zumkeller_, Aug 12 2009
%C A054531 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 A054531 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 A054531 Reinhard Zumkeller, <a href="/A054531/b054531.txt">Rows n = 1..120 of triangle, flattened</a>
%H A054531 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 A054531 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 A054531 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 A054531 Triangle begins
%e A054531    1;
%e A054531    2,  1;
%e A054531    3,  3,  1;
%e A054531    4,  2,  4,  1;
%e A054531    5,  5,  5,  5,  1;
%e A054531    6,  3,  2,  3,  6,  1;
%e A054531    7,  7,  7,  7,  7,  7,  1;
%e A054531    8,  4,  8,  2,  8,  4,  8,  1;
%e A054531    9,  9,  3,  9,  9,  3,  9,  9,  1;
%e A054531   10,  5, 10,  5,  2,  5, 10,  5, 10,  1;
%e A054531   11, 11, 11, 11, 11, 11, 11, 11, 11, 11,  1;
%e A054531   12,  6,  4,  3, 12,  2, 12,  3,  4,  6, 12,  1;
%e A054531   13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13,  1;
%t A054531 Table[n/GCD[n,k], {n,1,10}, {k,1,n}]//Flatten (* _G. C. Greubel_, Sep 13 2017 *)
%o A054531 (Haskell)
%o A054531 a054531 n k = div n $ gcd n k
%o A054531 a054531_row n = a054531_tabl !! (n-1)
%o A054531 a054531_tabl = zipWith (\u vs -> map (div u) vs) [1..] a050873_tabl
%o A054531 -- _Reinhard Zumkeller_, Jun 10 2013
%o A054531 (PARI) for(n=1,10, for(k=1,n, print1(n/gcd(n,k), ", "))) \\ _G. C. Greubel_, Sep 13 2017
%Y A054531 Cf. A050873, A164306, A226314, A277227 (row reversed, k=0..n-1).
%K A054531 nonn,tabl,frac,easy
%O A054531 1,2
%A A054531 _N. J. A. Sloane_, Apr 09 2000