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Square array read by antidiagonals: rows are successively recursivized versions of Bulgarian solitaire operation (starting from the usual "first order" version, A243051), as applied to partitions listed in A241918.
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%I #13 Apr 01 2017 20:56:31

%S 1,2,1,4,2,1,3,4,2,1,8,3,4,2,1,25,8,3,4,2,1,16,6,8,3,4,2,1,9,16,6,8,3,

%T 4,2,1,9,5,16,6,8,3,4,2,1,343,9,5,16,6,8,3,4,2,1,32,12,9,5,16,6,8,3,4,

%U 2,1,10,32,12,9,5,16,6,8,3,4,2,1,64,35,32,12,9,5,16,6,8,3,4,2,1,14641,64,10,32,12,9,5,16,6,8,3,4,2,1,125,24,64,10,32,12,9,5,16,6,8,3,4,2,1

%N Square array read by antidiagonals: rows are successively recursivized versions of Bulgarian solitaire operation (starting from the usual "first order" version, A243051), as applied to partitions listed in A241918.

%C The array is read by antidiagonals: A(0,0), A(0,1), A(1,0), A(0,2), A(1,1), A(2,0), ... .

%C Each row is a A241909-conjugate of the corresponding row in A243070.

%C Rows in both arrays converge towards A122111.

%C First point where row k differs from row k of A243070 seems to be A000040(k+2): primes from five onward: 5, 7, 11, 13, 17, 19, 23, 29, 31, ...

%C While the first point where row k differs from A122111 seems to begin as 6, 12, 24, 48, 96, 192, 384, 768, 1536, 3072, ... A007283 (3*2^n) from the term n=1 onward.

%C The rows in this table preserve A243503.

%e The top left corner of the array:

%e 1, 2, 4, 3, 8, 25, 16, 9, 9, 343, 32, 10, 64, 14641, 125, 27, ...

%e 1, 2, 4, 3, 8, 6, 16, 5, 9, 12, 32, 35, 64, 24, 18, 25, ...

%e 1, 2, 4, 3, 8, 6, 16, 5, 9, 12, 32, 10, 64, 24, 18, 7, ...

%e 1, 2, 4, 3, 8, 6, 16, 5, 9, 12, 32, 10, 64, 24, 18, 7, ...

%e 1, 2, 4, 3, 8, 6, 16, 5, 9, 12, 32, 10, 64, 24, 18, 7, ...

%o (Scheme)

%o (define (A243060 n) (A243060bi (A002260 n) (A004736 n)))

%o (define (A243060bi row col) (explist->n (ascpart_to_prime-exps (bulgarian-operation-n-th-order (prime-exps_to_ascpart (primefacs->explist col)) row))))

%o (define (bulgarian-operation-n-th-order ascpart n) (if (or (zero? n) (null? ascpart)) ascpart (let ((newpart (length ascpart))) (let loop ((newpartition (list)) (ascpart ascpart)) (cond ((null? ascpart) (sort (cons newpart (bulgarian-operation-n-th-order newpartition (- n 1))) <)) (else (loop (if (= 1 (car ascpart)) newpartition (cons (- (car ascpart) 1) newpartition)) (cdr ascpart))))))))

%o ;; For other required functions and libraries, please see A243051.

%Y Row 1: A243051, Row 2: A243052, Row 3: A243053.

%Y Rows converge towards A122111.

%Y Cf. A243070, A243503.

%K nonn,tabl

%O 1,2

%A _Antti Karttunen_, May 29 2014