A225632 Irregular table read by rows: n-th row gives distinct values of successively iterated Landau-like functions for n, starting with the initial value 1.

1, 1, 2, 1, 3, 6, 1, 4, 12, 1, 6, 30, 60, 1, 6, 30, 60, 1, 12, 84, 420, 1, 15, 120, 840, 1, 20, 180, 1260, 2520, 1, 30, 210, 840, 2520, 1, 30, 420, 4620, 13860, 27720, 1, 60, 660, 4620, 13860, 27720, 1, 60, 780, 8580, 60060, 180180, 360360

Offset

1,3

Comments

The leftmost column of table (the initial term of each row, T(n,1)) is 1, corresponding to lcm(1,1,...,1) computed from the {1+1+...+1} partition of n, after which, on the same row, each further term T(n,i) is computed by finding such a partition [p1,p2,...,pk] of n so that value of lcm(T(n,i-1), p1,p2,...,pk) is maximized, until finally A003418(n) is reached, which will be listed as the last term of row n (as the result would not change after that, if we continued the same process).

Links

Alois P. Heinz, Rows n = 1..150, flattened

Index entries for sequences related to lcm's

Example

The first fifteen rows of table are:
  1;
  1,   2;
  1,   3,    6;
  1,   4,   12;
  1,   6,   30,    60;
  1,   6,   30,    60;
  1,  12,   84,   420;
  1,  15,  120,   840;
  1,  20,  180,  1260,   2520;
  1,  30,  210,   840,   2520;
  1,  30,  420,  4620,  13860,  27720;
  1,  60,  660,  4620,  13860,  27720;
  1,  60,  780,  8580,  60060, 180180, 360360;
  1,  84, 1260, 16380, 180180, 360360;
  1, 105, 4620, 60060, 180180, 360360;

Maple

b:= proc(n, i) option remember; `if`(n=0, {1},
      `if`(i<1, {}, {seq(map(x->ilcm(x, `if`(j=0, 1, i)),
       b(n-i*j, i-1))[], j=0..n/i)}))
    end:
T:= proc(n) option remember; local d, h, l, ll;
      l:= b(n$2); ll:= NULL; d:=1; h:=0;
      while d<>h do ll:= ll, d; h:= d;
        d:= max(seq(ilcm(h, i), i=l))
      od; ll
    end:
seq(T(n), n=1..20);  # Alois P. Heinz, May 29 2013

Mathematica

b[n_, i_] := b[n, i] = If[n==0, {1}, If[i<1, {}, Table[Map[Function[{x}, LCM[x, If[j==0, 1, i]]], b[n-i*j, i-1]], {j, 0, n/i}]]]; T[n_] := T[n] = Module[{d, h, l, ll}, l=b[n, n]; ll={}; d=1; h=0; While[d != h, AppendTo[ll, d]; h=d; d = Max[ Table[LCM[h, i], {i, l}]]]; ll]; Table[T[n], {n, 1, 20}] // Flatten (* Jean-François Alcover, Jul 29 2015, after Alois P. Heinz *)

Prog

(Scheme) ;; With Antti Karttunen's IntSeq-library:
(definec (A225632 n) (A225630bi (Aux_for_225632 n) (- n (A225635 (Aux_for_225632 n))))) ;; Scheme-definition for A225630bi given in A225630.
(define Aux_for_225632 (COMPOSE -1+ (LEAST-GTE-I 1 1 A225635) 1+)) ;; Auxiliary function not submitted separately, which gives the row-number for the n-th term.
;; It starts as 1, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 9, ...

Crossrefs

Cf. A225634 (length of n-th row), A000793 (n>=2 gives the second column).

Cf. A225629 (second largest/rightmost term of n-th row).

Cf. A003418 (largest/rightmost term of n-th row).

Cf. A225630, A225631, A225635, A212721.

Cf. A225642 (row n starts from n instead of 1).

Cf. A226055 (the first term common with A225642 on the n-th row).

Cf. A225638 (distance to that first common term from the beginning of the row n).

Cf. A226056 (number of trailing terms common with A225642 on the n-th row).

Sequence in context: A103280 A046899 A309220 * A035206 A210238 A209936

Adjacent sequences: A225629 A225630 A225631 * A225633 A225634 A225635

Keyword

nonn,tabf

Author

Antti Karttunen, May 13 2013

Status

approved