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

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

Offset: 0

Author

Antti Karttunen, May 13 2013

Keywords

• nonn
• tabl

Comments

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.

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

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.

Examples

			Table begins:
  1, 1, 1, 1,  1,  1,  1,   1,   1,  1, ...
  1, 1, 2, 3,  4,  6,  6,  12,  15, 20, ...
  1, 1, 2, 6, 12, 30, 30,  84, 120, ...
  1, 1, 2, 6, 12, 60, 60, 420, 840, ...
  ...
		

Links

• Index entries for sequences related to lcm's

Crossrefs

Transpose: A225631. Cf. also A225632, A225634.

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

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

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

Programs

Scheme

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