A241909 -id:A241909 - cp's OEIS frontend

A243065 Permutation of natural numbers, the odd bisection of A241909 halved; equally, a composition of A064216 and A241909: a(n) = A241909(A064216(n)).

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

Offset: 1

Antti Karttunen, Jun 01 2014

nonn

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

Antti Karttunen, Table of n, a(n) for n = 1..512
Index entries for sequences that are permutations of the natural numbers

Inverse: A243066.

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

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

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.

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)).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jun 01 2014

nonn

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?

Antti Karttunen, Table of n, a(n) for n = 1..512
Index entries for sequences that are permutations of the natural numbers

Inverse: A243065.

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

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).

A243051 Integer sequence induced by Bulgarian solitaire operation on partition list A241918: a(n) = A241909(A242424(A241909(n))).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, May 29 2014

nonn

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.

			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.

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

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

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.

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].

A243062 Permutation of natural numbers, a composition of A048673 and A241909: a(n) = A241909(A048673(n)).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jun 02 2014

nonn

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

Antti Karttunen, Table of n, a(n) for n = 1..242
Index entries for sequences that are permutations of the natural numbers

Inverse permutation: A243061.

Cf. A048673, A241909, A243065-A243066.

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

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

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

A278220 Filtering sequence (related to prime factorization): a(n) = A046523(A241909(n)).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Nov 16 2016

nonn

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

Cf. A046523, A075158, A241909.

Cf. also A278219, A278221.

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

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

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

A331595 a(n) = gcd(A122111(n), A241909(n)).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jan 22 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences computed from indices in prime factorization

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

Cf. also A280489, A280491.

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 *)

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));

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

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

A243052 Integer sequence induced by second order Bulgarian solitaire operation on partition list A241918: a(n) = A241909(A243072(A241909(n))).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, May 29 2014

nonn

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.

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

Second row of A243060.

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

(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.

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

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

Original entry on oeis.org

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

Antti Karttunen, May 29 2014

nonn

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.

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

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.

(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.

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

A331299 a(n) = min(n, A241909(n)).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jan 17 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from indices in prime factorization

Cf. A241909.

Cf. also A331170, A331280.

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

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

A243061 Permutation of natural numbers, a composition of A241909 and A064216: a(n) = A064216(A241909(n)).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jun 02 2014

nonn

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

Inverse permutation: A243062.

Cf. A064216, A241909, A243065-A243066.

A064216(n) = A064989(n+n-1);
A064989(n) = { my(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) };
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));
A243061(n) = A064216(A241909(n)); \\ Antti Karttunen, Dec 10 2021

(define (A243061 n) (A064216 (A241909 n)))

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

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

cp's OEIS Frontend

A243065 Permutation of natural numbers, the odd bisection of A241909 halved; equally, a composition of A064216 and A241909: a(n) = A241909(A064216(n)).

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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)).

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A243051 Integer sequence induced by Bulgarian solitaire operation on partition list A241918: a(n) = A241909(A242424(A241909(n))).

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A243062 Permutation of natural numbers, a composition of A048673 and A241909: a(n) = A241909(A048673(n)).

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A278220 Filtering sequence (related to prime factorization): a(n) = A046523(A241909(n)).

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A331595 a(n) = gcd(A122111(n), A241909(n)).

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A243052 Integer sequence induced by second order Bulgarian solitaire operation on partition list A241918: a(n) = A241909(A243072(A241909(n))).

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

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A331299 a(n) = min(n, A241909(n)).

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