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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

%I

%S 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,

%T 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,

%U 840,420,60,60,12,6,2,1,1,11,210,1260,840,420,60,60,12,6,2,1,1

%N 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.

%C In this array the maximization of LCM starts from partition {k} of k, instead of partition {1+1+...+1} as in A225630.

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

%e The top-left corner of the array:

%e 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, ...

%e 1, 1, 2, 6, 12, 30, 30, 84, 120, 180, 210, 330, 420, ...

%e 1, 1, 2, 6, 12, 60, 60, 420, 840, 1260, 840, 4620, 4620, ...

%e 1, 1, 2, 6, 12, 60, 60, 420, 840, 2520, 2520, 13860, 13860, ...

%e 1, 1, 2, 6, 12, 60, 60, 420, 840, 2520, 2520, 27720, 27720, ...

%e ...

%o (Scheme):

%o (define (A225640 n) (A225640bi (A025581 n) (A002262 n)))

%o (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)))))))

%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: A225641.

%Y Cf. A225642, A225644, A001477 (row 0), A225646 (row 1).

%Y Rows converge towards A003418 (main diagonal of this array).

%Y 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.

%K nonn,tabl

%O 0,4

%A _Antti Karttunen_, May 14 2013

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Last modified June 24 10:02 EDT 2019. Contains 324323 sequences. (Running on oeis4.)