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

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, 96, 25, 40, 1024, 30, 2048, 49, 72, 192, 54, 21, 4096, 384, 144, 637, 8192, 60, 16384, 80, 50, 768, 32768, 22, 81, 45, 288, 160, 65536, 35, 108, 22627, 576, 1536, 131072

Offset: 1

Author

Antti Karttunen, May 29 2014

Keywords

• nonn

Comments

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.

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.

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.

How the partitions are encoded in this case, see A241918.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..1024

Crossrefs

Third row of A243060.

Differs from A122111 for the first time at n=24, where a(24) = 55, while A122111(24) = 14.

Cf. A243051, A243052, A241918, A241909, A243073.

Programs

Scheme

(define (A243053 n) (explist->n (ascpart_to_prime-exps (bulgarian-operation-n-th-order (prime-exps_to_ascpart (primefacs->explist n)) 3))))
(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))))))))
;; Other required functions and libraries, please see A243051.

Formula

a(n) = A241909(A243073(A241909(n))).