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A225640
Array A(n,k) of iterated Landau-like functions, where on the row n=0 A(0,0)=1 and A(0,k>=1)=k, and the successive rows A(n,k) give a maximum value lcm(p1,p2,...,pj,A(n-1,k)) for all partitions {p1+p2+...+pj} of k; square array A(n,k), n>=0, k>=0, read by antidiagonals.
14
1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 4, 6, 2, 1, 1, 5, 12, 6, 2, 1, 1, 6, 30, 12, 6, 2, 1, 1, 7, 30, 60, 12, 6, 2, 1, 1, 8, 84, 60, 60, 12, 6, 2, 1, 1, 9, 120, 420, 60, 60, 12, 6, 2, 1, 1, 10, 180, 840, 420, 60, 60, 12, 6, 2, 1, 1, 11, 210, 1260, 840, 420, 60, 60, 12, 6, 2, 1, 1
OFFSET
0,4
COMMENTS
In this array the maximization of LCM starts from partition {k} of k, instead of partition {1+1+...+1} as in A225630.
EXAMPLE
The top-left corner of the array:
1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, ...
1, 1, 2, 6, 12, 30, 30, 84, 120, 180, 210, 330, 420, ...
1, 1, 2, 6, 12, 60, 60, 420, 840, 1260, 840, 4620, 4620, ...
1, 1, 2, 6, 12, 60, 60, 420, 840, 2520, 2520, 13860, 13860, ...
1, 1, 2, 6, 12, 60, 60, 420, 840, 2520, 2520, 27720, 27720, ...
...
PROG
(Scheme):
(define (A225640 n) (A225640bi (A025581 n) (A002262 n)))
(define (A225640bi col row) (let ((maxlcm (list 0))) (let loop ((prevmaxlcm (max 1 col)) (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))))))))
CROSSREFS
Transpose: A225641.
Cf. A225642, A225644, A001477 (row 0), A225646 (row 1).
Rows converge towards A003418 (main diagonal of this array).
See A225630 for a variant employing a similar process, but which uses 1 in column n as the initial seed for that column, instead of n.
Sequence in context: A114282 A112739 A308813 * A345418 A194543 A287920
KEYWORD
nonn,tabl
AUTHOR
Antti Karttunen, May 14 2013
STATUS
approved