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A225632
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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.
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14
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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
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OFFSET
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1,3
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COMMENTS
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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).
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LINKS
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EXAMPLE
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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;
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MAPLE
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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:
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MATHEMATICA
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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 *)
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PROG
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(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, ...
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CROSSREFS
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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. A225642 (row n starts from n instead of 1).
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).
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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