A201144 -id:A201144 - cp's OEIS frontend

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

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, 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, 11, 11, 11, 1, 12, 6, 4, 3, 12, 2, 12, 3, 4, 6, 12, 1, 13, 13, 13, 13, 13

Offset: 1

N. J. A. Sloane, Apr 09 2000

Sum of n-th row = A057660(n). - Reinhard Zumkeller, Aug 12 2009
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
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

			Triangle begins
   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,  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, 11, 11, 11,  1;
  12,  6,  4,  3, 12,  2, 12,  3,  4,  6, 12,  1;
  13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13,  1;

Reinhard Zumkeller, Rows n = 1..120 of triangle, flattened
Lance Fortnow, Counting the Rationals Quickly, Computational Complexity Weblog, Monday, March 01, 2004.
R. J. Mathar, Plots of cycle graphs of the finite groups up to order 36, (2015).
Yoram Sagher, Counting the rationals, Amer. Math. Monthly, 96 (1989), p. 823. Math. Rev. 90i:04001.

Cf. A050873, A164306, A226314, A277227 (row reversed, k=0..n-1).

a054531 n k = div n $ gcd n k
a054531_row n = a054531_tabl !! (n-1)
a054531_tabl = zipWith (\u vs -> map (div u) vs) [1..] a050873_tabl
-- Reinhard Zumkeller, Jun 10 2013

Table[n/GCD[n,k], {n,1,10}, {k,1,n}]//Flatten (* G. C. Greubel, Sep 13 2017 *)

for(n=1,10, for(k=1,n, print1(n/gcd(n,k), ", "))) \\ G. C. Greubel, Sep 13 2017

A183110 Period-length of the ultimate periodic behavior of the orbit of a list [1,1,1,...,1] of n 1's under the mapping defined in the comments.

Original entry on oeis.org

1, 2, 1, 3, 3, 1, 4, 4, 4, 1, 5, 5, 5, 5, 1, 6, 6, 6, 6, 6, 1, 7, 7, 7, 7, 7, 7, 1, 8, 8, 8, 8, 8, 8, 8, 1, 9, 9, 9, 9, 9, 9, 9, 9, 1, 10, 10, 10, 10, 10, 10, 10, 10, 10, 1, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 1, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 1, 13, 13

Offset: 1

John W. Layman, Feb 01 2011

We use the list mapping introduced in A092964, whereby one removes the first term of the list, z(1), and adds 1 to each of the next z(1) terms (appending 1's if necessary) to get a new list.

This is also conjectured to be the length of the longest cycle of pebble-moves among the partitions of n (cf. A201144). - Andrew V. Sutherland

			Under the indicated mapping the list [1,1,1,1,1,1,1] of seven 1's results in the orbit [1,1,1,1,1,1,1], [2,1,1,1,1,1], [2,2,1,1,1], [3,2,1,1], [3,2,2], [3,3,1], [4,2,1], [3,2,1,1], ... which is clearly periodic with period-length 4, so a(7) = 4.

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A037306, A092964, A178572, A178574.

f[p_] := Module[{pp, x, lpp, m, i}, pp = p; x = pp[[1]]; pp = Delete[pp,1]; lpp = Length[pp]; m = Min[x, lpp]; For[i = 1, i ≤ m, i++, pp[[i]]++]; For[i = 1, i ≤ x - lpp, i++, AppendTo[pp, 1]]; pp]; orb[p_] := Module[{s, v}, v = p; s = {v}; While[! MemberQ[s, v = f[v]], AppendTo[s, v]]; s]; attractor[p_] := Module[{orbp, pos, len, per}, orbp = orb[p]; pos = Flatten[Position[orbp, f[orbp[[-1]]]]][[1]] - 1; (*pos = steps to enter period*) len = Length[orbp] - pos; per = Take[orbp, -len]; Sort[per]]; a = {}; For[n = 1, n ≤ 80, n++, {rn = Table[1, {k, 1, n}]; orbn = orb[rn]; lenorb = Length[orbn]; lenattr = Length[attractor[rn]]; AppendTo[a, lenattr]}]; Print[a];

It appears, but has not yet been proved, that a(n)=1 if n=t(k) and a(n)=k if t(k-1) < n < t(k) where t(k) is the k-th triangular number t(k) = k*(k+1)/2.

cp's OEIS Frontend

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

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