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A243053 Integer sequence induced by third-order Bulgarian solitaire operation on partition list A241918: a(n) = A241909(A243073(A241909(n))).

%I A243053 #17 Aug 03 2023 01:37:02
%S A243053 1,2,4,3,8,6,16,5,9,12,32,10,64,24,18,7,128,15,256,20,36,48,512,55,27,
%T A243053 96,25,40,1024,30,2048,49,72,192,54,21,4096,384,144,637,8192,60,16384,
%U A243053 80,50,768,32768,22,81,45,288,160,65536,35,108,22627,576,1536,131072
%N A243053 Integer sequence induced by third-order Bulgarian solitaire operation on partition list A241918: a(n) = A241909(A243073(A241909(n))).
%C A243053 The usual (first-order) Bulgarian Solitaire operation (cf. A243051) applied to an unordered integer partition means: subtract one from each part, and add a new part as large as there were parts in the old partition.
%C A243053 The "Second-Order Bulgarian Operation" means that after subtracting one from each part of the old partition (and discarding the parts that diminished to zero), we apply the (first-order) Bulgarian operation to the remaining partition before adding a new part as large as there were parts in the original partition.
%C A243053 Similarly, in "Third-Order Bulgarian Solitaire Operation", we apply the Second-Order Bulgarian operation to the remaining partition (after we have subtracted one from each part) before adding a new part as large as there were parts in the original partition.
%C A243053 How the partitions are encoded in this case, see A241918.
%H A243053 Antti Karttunen, <a href="/A243053/b243053.txt">Table of n, a(n) for n = 1..1024</a>
%F A243053 a(n) = A241909(A243073(A241909(n))).
%o A243053 (Scheme)
%o A243053 (define (A243053 n) (explist->n (ascpart_to_prime-exps (bulgarian-operation-n-th-order (prime-exps_to_ascpart (primefacs->explist n)) 3))))
%o A243053 (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 A243053 ;; Other required functions and libraries, please see A243051.
%Y A243053 Third row of A243060.
%Y A243053 Differs from A122111 for the first time at n=24, where a(24) = 55, while A122111(24) = 14.
%Y A243053 Cf. A243051, A243052, A241918, A241909, A243073.
%K A243053 nonn
%O A243053 1,2
%A A243053 _Antti Karttunen_, May 29 2014