login
Array of iterated Landau-like functions, starting maximization of LCM from the partition {1+1+...+1} of n, read downwards antidiagonals.
16

%I #15 Apr 01 2017 20:44:05

%S 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 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 60,12,6,2,1,1,1,30,180,840,420,60,60,12,6,2,1,1

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

%C 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 This will produce the ordinary Landau's function (A000793) for row 1, the "second order Landau's function" (A225627) for row 2, etc.

%C 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 <a href="/index/Lc#lcm">Index entries for sequences related to lcm's</a>

%e Table begins:

%e 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...

%e 1, 1, 2, 3, 4, 6, 6, 12, 15, 20, ...

%e 1, 1, 2, 6, 12, 30, 30, 84, 120, ...

%e 1, 1, 2, 6, 12, 60, 60, 420, 840, ...

%e ...

%o (Scheme):

%o (define (A225630 n) (A225630bi (A025581 n) (A002262 n)))

%o (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 (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 Transpose: A225631. Cf. also A225632, A225634.

%Y Row 0: A000012, row 1: A000793, row 2: A225627, row 3: A225628. Cf. also A225629.

%Y Rows converge towards A003418, which is also the main diagonal of this array.

%Y See A225640 for a variant which uses a similar process, but where the "initial seed" in column n is n instead of 1.

%K nonn,tabl

%O 0,8

%A _Antti Karttunen_, May 13 2013