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A225630 Array of iterated Landau-like functions, starting maximization of LCM from the partition {1+1+...+1} of n, read downwards antidiagonals.

%I A225630 #18 Mar 05 2025 13:43:55
%S A225630 1,1,1,1,1,1,1,2,1,1,1,3,2,1,1,1,4,6,2,1,1,1,6,12,6,2,1,1,1,6,30,12,6,
%T A225630 2,1,1,1,12,30,60,12,6,2,1,1,1,15,84,60,60,12,6,2,1,1,1,20,120,420,60,
%U A225630 60,12,6,2,1,1,1,30,180,840,420,60,60,12,6,2,1,1
%N A225630 Array of iterated Landau-like functions, starting maximization of LCM from the partition {1+1+...+1} of n, read downwards antidiagonals.
%C A225630 Row 0 consists of all 1's (corresponding to lcm(1,1,...,1) computed from the {1+1+...+1} partition), after which, on each succeeding row, the entry A(row,n) is computed by finding such a partition {p1+p2+...+pk} of n that value of lcm(A(row-1,n),p1,p2,...,pk) is maximized.
%C A225630 This will produce the ordinary Landau's function (A000793) for row 1, the "second order Landau's function" (A225627) for row 2, etc.
%C A225630 For each column n, only finite number of distinct values (A225634(n)) occur, after which the fixed point A003418(n) that has been reached repeats forever.
%H A225630 <a href="/index/Lc#lcm">Index entries for sequences related to lcm's</a>
%e A225630 Table begins:
%e A225630   1, 1, 1, 1,  1,  1,  1,   1,   1,  1, ...
%e A225630   1, 1, 2, 3,  4,  6,  6,  12,  15, 20, ...
%e A225630   1, 1, 2, 6, 12, 30, 30,  84, 120, ...
%e A225630   1, 1, 2, 6, 12, 60, 60, 420, 840, ...
%e A225630   ...
%o A225630 (Scheme)
%o A225630 (define (A225630 n) (A225630bi (A025581 n) (A002262 n)))
%o A225630 (define (A225630bi col row) (let ((maxlcm (list 0))) (let loop ((prevmaxlcm 1) (stepsleft row)) (if (zero? stepsleft) prevmaxlcm (begin (gen_partitions col (lambda (p) (set-car! maxlcm (max (car maxlcm) (apply lcm (cons prevmaxlcm p)))))) (loop (car maxlcm) (- stepsleft 1)))))))
%o A225630 (define (gen_partitions m colfun) (let recurse ((m m) (b m) (n 0) (partition (list))) (cond ((zero? m) (colfun partition)) (else (let loop ((i 1)) (recurse (- m i) i (+ 1 n) (cons i partition)) (if (< i (min b m)) (loop (+ 1 i))))))))
%Y A225630 Transpose: A225631. Cf. also A225632, A225634.
%Y A225630 Row 0: A000012, row 1: A000793, row 2: A225627, row 3: A225628. Cf. also A225629.
%Y A225630 Rows converge towards A003418, which is also the main diagonal of this array.
%Y A225630 See A225640 for a variant which uses a similar process, but where the "initial seed" in column n is n instead of 1.
%K A225630 nonn,tabl
%O A225630 0,8
%A A225630 _Antti Karttunen_, May 13 2013