A241909 -id:A241909

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A243065

Permutation of natural numbers, the odd bisection of A241909 halved; equally, a composition of A064216 and A241909: a(n) = A241909(A064216(n)).

+20
18

1, 2, 4, 8, 3, 16, 32, 9, 64, 128, 27, 256, 6, 5, 512, 1024, 81, 18, 2048, 243, 4096, 8192, 25, 16384, 12, 729, 32768, 54, 2187, 65536, 131072, 125, 162, 262144, 6561, 524288, 1048576, 15, 36, 2097152, 7, 4194304, 486, 19683, 8388608, 108, 59049, 1458, 16777216, 625, 33554432, 67108864, 75

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Are there any other fixed points than 1, 2, 18 and 72?

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..512

Index entries for sequences that are permutations of the natural numbers

FORMULA

a(1) = 1, and for n>=2, a(n) = A241909(2n-1)/2. Equally, a(n) = ceiling(A241909(2n-1)/2) for all n.

As a composition of related permutations:

a(n) = A241909(A064216(n)).

a(n) = A241909(A243061(A241909(n))).

For all n, a(A006254(n)) = 2^n.

PROG

(Scheme) (define (A243065 n) (A241909 (A064216 n)))

CROSSREFS

Inverse: A243066.

Cf. A064216, A241909, A243505-A243506, A244152-A244154, A243061-A243062, A006254.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Jun 01 2014

STATUS

approved

A243066

Permutation of natural numbers, the even bisection of A241909 incremented by one and halved; equally, a composition of A241909 and A048673: a(n) = A048673(A241909(n)).

+20
18

1, 2, 5, 3, 14, 13, 41, 4, 8, 63, 122, 25, 365, 313, 38, 6, 1094, 18, 3281, 172, 188, 1563, 9842, 61, 23, 7813, 11, 1201, 29525, 123, 88574, 7, 938, 39063, 113, 39, 265721, 195313, 4688, 666, 797162, 858, 2391485, 8404, 74, 976563, 7174454, 85, 68, 88, 23438, 58825, 21523361, 28

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

For n > 1, 2n is found in A241909 from the position (2*a(n))-1. I.e., A241909((2*a(n))-1) = 2n for all n >= 2.

Or in other words, a(n) gives the position in the odd bisection of A241909 where 2n is located at.

Are there any other fixed points than 1, 2, 18 and 72?

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..512

Index entries for sequences that are permutations of the natural numbers

FORMULA

a(1) = 1, a(n) = (A241909(2*n)+1)/2.

As a composition of related permutations:

a(n) = A048673(A241909(n)).

a(n) = A241909(A243062(A241909(n))).

For all n>=1, a(2^n) = A006254(n).

PROG

(Scheme, two alternatives)

(define (A243066 n) (if (= n 1) 1 (/ (+ 1 (A241909 (* 2 n))) 2)))

(define (A243066 n) (A048673 (A241909 n)))

CROSSREFS

Inverse: A243065.

Cf. A048673, A241909, A243505-A243506, A244152-A244154, A243061-A243062.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Jun 01 2014

STATUS

approved

A243051

Integer sequence induced by Bulgarian solitaire operation on partition list A241918: a(n) = A241909(A242424(A241909(n))).

+20
9

1, 2, 4, 3, 8, 25, 16, 9, 9, 343, 32, 10, 64, 14641, 125, 27, 128, 15, 256, 98, 2401, 371293, 512, 30, 27, 24137569, 6, 2662, 1024, 147, 2048, 81, 161051, 893871739, 625, 50, 4096, 78310985281, 4826809, 28, 8192, 3993, 16384, 57122, 50, 14507145975869, 32768, 90, 81

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

In "Bulgarian solitaire" a deck of cards or another finite set of objects is divided into one or more piles, and the "Bulgarian operation" is performed by taking one card from each pile, and making a new pile of them, which is added to the remaining set of piles. Essentially, this operation is a function whose domain and range are unordered integer partitions (cf. A000041) and which preserves the total size of a partition (the sum of its parts). This sequence is induced when the operation is implemented on the partitions as ordered by the list A241918.

REFERENCES

Martin Gardner, Colossal Book of Mathematics, Chapter 34, Bulgarian Solitaire and Other Seemingly Endless Tasks, pp. 455-467, W. W. Norton & Company, 2001.

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..256

Ethan Akin and Morton Davis, "Bulgarian solitaire", American Mathematical Monthly 92 (4): 237-250. (1985).

Wikipedia, Bulgarian solitaire

FORMULA

a(n) = A241909(A242424(A241909(n))).

a(n) = 1 + A075157(A226062(A075158(n-1))).

A243503(a(n)) = A243503(n) for all n. [Because Bulgarian operation doesn't change the total sum of the partition].

EXAMPLE

For n = 10, we see that as 10 = 2*5 = p_1^1 * p_2^0 * p_3^1, it encodes a partition [2,2,2]. Applying one step of Bulgarian solitaire (subtract one from each part, and add a new part as large as there were parts in the old partition) to this partition results a new partition [1,1,1,3], which is encoded in the prime factorization of p_1^0 * p_2^0 * p_3^0 * p_4^3 = 7^3 = 343. Thus a(10) = 343.

For n = 46, we see that as 46 = 2*23 = p_1 * p_9 = p_1^1 * p_2^0 * p_3^0 * ... * p_9^1, it encodes a partition [2,2,2,2,2,2,2,2,2]. Applying one step of Bulgarian solitaire to this partition results a new partition [1,1,1,1,1,1,1,1,1,9], which is encoded in the prime factorization of p_1^0 * p_2^0 * ... * p_9^0 * p_10^9 = 29^9 = 14507145975869. Thus a(46) = 14507145975869.

For n = 1875, we see that as 1875 = p_1^0 * p_2^1 * p_3^4, it encodes a partition [1,2,5]. Applying Bulgarian Solitaire, we get a new partition [1,3,4]. This in turn is encoded by p_1^0 * p_2^2 * p_3^2 = 3^2 * 5^2 = 225. Thus a(1875)=225.

PROG

(Scheme, three different implementations)

(define (A243051 n) (A241909 (A242424 (A241909 n))))

(define (A243051 n) (1+ (A075157 (A226062 (A075158 (- n 1))))))

;; The following requires Aubrey Jaffer's SLIB Scheme library:

(require 'factor)

(define (A243051 n) (explist->n (ascpart_to_prime-exps (bulgarian-operation (prime-exps_to_ascpart (primefacs->explist n))))))

(define (bulgarian-operation ascpart) (let loop ((newpartition (list (length ascpart))) (ascpart ascpart)) (cond ((null? ascpart) (sort newpartition <)) (else (loop (if (= 1 (car ascpart)) newpartition (cons (- (car ascpart) 1) newpartition)) (cdr ascpart))))))

(define (primefacs->explist n) (reverse! (primefactorization->explist n)))

(define (primefactorization->explist n) (if (= 1 n) (list) (let loop ((factors (sort (factor n) <)) (pf 1) (el (list))) (cond ((null? factors) el) ((= (car factors) pf) (set-car! el (1+ (car el))) (loop (cdr factors) (car factors) el)) (else (loop (cdr factors) (car factors) (cons 1 (cons-n-times (-1+ (- (A049084 (car factors)) (A049084 pf))) 0 el))))))))

(define (prime-exps_to_ascpart explist) (if (null? explist) explist (sub1from_the_last (partsums (cons (+ (car explist) 1) (cdr explist))))))

(define (partsums a) (cdr (reverse! (fold-left (lambda (psums n) (cons (+ n (car psums)) psums)) (list 0) a))))

(define (sub1from_the_last lista) (let ((rev (reverse lista))) (reverse! (cons (- (car rev) 1) (cdr rev)))))

(define (ascpart_to_prime-exps partlist) (if (null? partlist) partlist (add1to_the_last (cons (- (car partlist) 1) (diff partlist)))))

(define (add1to_the_last lista) (let ((rev (reverse lista))) (reverse! (cons (+ 1 (car rev)) (cdr rev)))))

(define (diff a) (map - (cdr a) (reverse! (cdr (reverse a)))))

(define (explist->n explist) (if (null? explist) 1 (mul (lambda (i) (expt (A000040 i) (list-ref explist (-1+ i)))) 1 (length explist))))

CROSSREFS

Row 1 of A243060 (table which gives successive "recursive iterates" of this sequence and converges towards A122111).

Fixed points: A243054.

Cf. A243052, A243053, A243503, A242424, A241909, A226062, A075157, A075158.

KEYWORD

nonn

AUTHOR

Antti Karttunen, May 29 2014

STATUS

approved

A243062

Permutation of natural numbers, a composition of A048673 and A241909: a(n) = A241909(A048673(n)).

+20
7

1, 2, 4, 8, 3, 5, 9, 81, 64, 32, 16, 512, 6, 128, 15, 8192, 27, 6561, 25, 11, 625, 125, 18, 78125, 12, 729, 250, 45, 7, 65536, 256, 387420489, 162, 1024, 486, 1073741824, 54, 36, 16384, 2916, 243, 8388608, 49, 131072, 16807, 3125, 10, 17496, 262144, 531441, 121, 72, 75

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

This is A241909-conjugate of A243066. Please see the comments at the latter sequence.

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..242

Index entries for sequences that are permutations of the natural numbers

FORMULA

a(n) = A241909(A048673(n)).

a(n) = A241909(A243066(A241909(n))).

PROG

(Scheme) (define (A243062 n) (A241909 (A048673 n)))

CROSSREFS

Inverse permutation: A243061.

Cf. A048673, A241909, A243065-A243066.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Jun 02 2014

STATUS

approved

A278220

Filtering sequence (related to prime factorization): a(n) = A046523(A241909(n)).

+20
7

1, 2, 4, 2, 8, 4, 16, 2, 6, 8, 32, 4, 64, 16, 12, 2, 128, 6, 256, 8, 24, 32, 512, 4, 12, 64, 6, 16, 1024, 12, 2048, 2, 48, 128, 36, 6, 4096, 256, 96, 8, 8192, 24, 16384, 32, 12, 512, 32768, 4, 24, 12, 192, 64, 65536, 6, 72, 16, 384, 1024, 131072, 12, 262144, 2048, 24, 2, 144, 48, 524288, 128, 768, 36, 1048576, 6, 2097152, 4096, 30, 256, 72, 96, 4194304, 8, 6, 8192

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..410

FORMULA

a(n) = A046523(A241909(n)).

a(n) = A278219(A075158(n-1)).

PROG

(Scheme) (define (A278220 n) (A046523 (A241909 n)))

CROSSREFS

Cf. A046523, A075158, A241909.

Cf. also A278219, A278221.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Nov 16 2016

STATUS

approved

A331595

a(n) = gcd(A122111(n), A241909(n)).

+20
7

1, 2, 4, 3, 8, 3, 16, 5, 3, 3, 32, 5, 64, 3, 18, 7, 128, 15, 256, 5, 18, 3, 512, 7, 3, 3, 5, 5, 1024, 15, 2048, 11, 18, 3, 18, 7, 4096, 3, 18, 7, 8192, 15, 16384, 5, 50, 3, 32768, 11, 3, 45, 18, 5, 65536, 7, 108, 7, 18, 3, 131072, 7, 262144, 3, 50, 13, 108, 15, 524288, 5, 18, 45, 1048576, 11, 2097152, 3, 15, 5, 18, 15, 4194304, 11, 7, 3

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..16384

Index entries for sequences computed from indices in prime factorization

FORMULA

a(n) = gcd(A122111(n), A241909(n)).

a(A241916(n)) = a(n).

MATHEMATICA

Array[If[# == 1, 1, GCD @@ {Block[{k = #, m = 0}, Times @@ Power @@@ Table[k -= m; k = DeleteCases[k, 0]; {Prime@ Length@ k, m = Min@ k}, Length@ Union@ k]] &@ Catenate[ConstantArray[PrimePi[#1], #2] & @@@ #], Function[t, Times @@ Prime@ Accumulate[If[Length@ t < 2, {0}, Join[{1}, ConstantArray[0, Length@ t - 2], {-1}]] + ReplacePart[t, Map[#1 -> #2 & @@ # &, #]]]]@ ConstantArray[0, Transpose[#][[1, -1]]] &[# /. {p_, e_} /; p > 0 :> {PrimePi@ p, e}]} &@ FactorInteger[#]] &, 82] (* Michael De Vlieger, Jan 24 2020, after JungHwan Min at A122111 *)

PROG

(PARI)

A064989(n) = {my(f); f = factor(n); if((n>1 && f[1, 1]==2), f[1, 2] = 0); for (i=1, #f~, f[i, 1] = precprime(f[i, 1]-1)); factorback(f)};

A122111(n) = if(1==n, n, prime(bigomega(n))*A122111(A064989(n)));

A241909(n) = if(1==n||isprime(n), 2^primepi(n), my(f=factor(n), h=1, i, m=1, p=1, k=1); while(k<=#f~, p = nextprime(1+p); i = primepi(f[k, 1]); m *= p^(i-h); h = i; if(f[k, 2]>1, f[k, 2]--, k++)); (p*m));

A331595(n) = gcd(A122111(n), A241909(n));

CROSSREFS

Cf. A122111, A241909, A241916, A331596 (number of distinct prime factors), A331597, A331598, A331599, A331600.

Cf. also A280489, A280491.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Jan 22 2020

STATUS

approved

A243052

Integer sequence induced by second order Bulgarian solitaire operation on partition list A241918: a(n) = A241909(A243072(A241909(n))).

+20
6

1, 2, 4, 3, 8, 6, 16, 5, 9, 12, 32, 35, 64, 24, 18, 25, 128, 15, 256, 539, 36, 48, 512, 14, 27, 96, 25, 17303, 1024, 175, 2048, 125, 72, 192, 54, 21, 4096, 384, 144, 154, 8192, 3773, 16384, 485537, 245, 768, 32768, 70, 81, 45, 288, 26977283, 65536, 10, 108, 3146, 576, 1536, 131072

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

The usual Bulgarian Solitaire operation (the "first order" version, 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 Solitaire" 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.

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

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..512

FORMULA

a(n) = A241909(A243072(A241909(n))).

PROG

(Scheme)

(define (A243052 n) (explist->n (ascpart_to_prime-exps (bulgarian-operation-n-th-order (prime-exps_to_ascpart (primefacs->explist n)) 2))))

(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.

CROSSREFS

Second row of A243060.

Cf. A243051, A243053, A241918, A241909, A243072.

KEYWORD

nonn

AUTHOR

Antti Karttunen, May 29 2014

STATUS

approved

A243053

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

+20
6

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

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

FORMULA

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

PROG

(Scheme)

(define (A243053 n) (explist->n (ascpart_to_prime-exps (bulgarian-operation-n-th-order (prime-exps_to_ascpart (primefacs->explist n)) 3))))

;; Other required functions and libraries, please see A243051.

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.

KEYWORD

nonn

AUTHOR

Antti Karttunen, May 29 2014

STATUS

approved

A331299

a(n) = min(n, A241909(n)).

+20
5

1, 2, 3, 3, 5, 6, 7, 5, 6, 10, 11, 12, 13, 14, 15, 7, 17, 15, 19, 20, 21, 22, 23, 24, 12, 26, 10, 28, 29, 30, 31, 11, 33, 34, 35, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 24, 45, 51, 52, 53, 21, 55, 56, 57, 58, 59, 60, 61, 62, 63, 13, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 30, 76, 72, 78, 79, 80, 14, 82

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..65537

Index entries for sequences computed from indices in prime factorization

FORMULA

a(n) = min(n, A241909(n)).

PROG

(PARI)

A331299(n) = min(n, A241909(n));

CROSSREFS

Cf. A241909.

Cf. also A331170, A331280.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Jan 17 2020

STATUS

approved

A243061

Permutation of natural numbers, a composition of A241909 and A064216: a(n) = A064216(A241909(n)).

+20
4

1, 2, 5, 3, 6, 13, 29, 4, 7, 47, 20, 25, 113, 95, 15, 11, 78, 23, 355, 158, 103, 267, 406, 89, 19, 1247, 17, 1237, 1577, 139, 660, 10, 221, 4363, 67, 38, 8179, 13109, 967, 393, 9266, 515, 21605, 4162, 28, 23601, 19578, 239, 43, 83, 987, 31247

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

This is A241909-conjugate of A243065. Please see the comments at the latter sequence.

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..1603

Index entries for sequences that are permutations of the natural numbers

Index entries for sequences computed from indices in prime factorization

FORMULA

a(n) = A064216(A241909(n)).

a(n) = A241909(A243065(A241909(n))).