A241909 -id:A241909

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A242378

Square array read by antidiagonals: to obtain A(i,j), replace each prime factor prime(k) in prime factorization of j with prime(k+i).

+10
17

0, 1, 0, 2, 1, 0, 3, 3, 1, 0, 4, 5, 5, 1, 0, 5, 9, 7, 7, 1, 0, 6, 7, 25, 11, 11, 1, 0, 7, 15, 11, 49, 13, 13, 1, 0, 8, 11, 35, 13, 121, 17, 17, 1, 0, 9, 27, 13, 77, 17, 169, 19, 19, 1, 0, 10, 25, 125, 17, 143, 19, 289, 23, 23, 1, 0, 11, 21, 49, 343, 19, 221, 23, 361, 29, 29, 1, 0

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

OFFSET

0,4

COMMENTS

Each row k is a multiplicative function, being in essence "the k-th power" of A003961, i.e., A(row,col) = A003961^row (col). Zeroth power gives an identity function, A001477, which occurs as the row zero.

The terms in the same column have the same prime signature.

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

LINKS

Antti Karttunen, Table of n, a(n) for n = 0..10439; Antidiagonals n = 0..143, flattened

FORMULA

A(0,n) = n, A(row,0) = 0, A(row>0,n>0) = A003961(A(row-1,n)).

EXAMPLE

The top-left corner of the array:

0, 1, 2, 3, 4, 5, 6, 7, 8, ...

0, 1, 3, 5, 9, 7, 15, 11, 27, ...

0, 1, 5, 7, 25, 11, 35, 13, 125, ...

0, 1, 7, 11, 49, 13, 77, 17, 343, ...

0, 1, 11, 13, 121, 17, 143, 19,1331, ...

0, 1, 13, 17, 169, 19, 221, 23,2197, ...

...

A(2,6) = A003961(A003961(6)) = p_{1+2} * p_{2+2} = p_3 * p_4 = 5 * 7 = 35, because 6 = 2*3 = p_1 * p_2.

PROG

(Scheme, with function factor from with Aubrey Jaffer's SLIB Scheme library)

(require 'factor)

(define (ifactor n) (cond ((< n 2) (list)) (else (sort (factor n) <))))

(define (A242378 n) (A242378bi (A002262 n) (A025581 n)))

(define (A242378bi row col) (if (zero? col) col (apply * (map A000040 (map (lambda (k) (+ k row)) (map A049084 (ifactor col)))))))

CROSSREFS

Taking every second column from column 2 onward gives array A246278 which is a permutation of natural numbers larger than 1.

Transpose: A242379.

Row 0: A001477, Row 1: A003961 (from 1 onward), Row 2: A357852 (from 1 onward), Row 3: A045968 (from 7 onward), Row 4: A045970 (from 11 onward).

Column 2: A000040 (primes), Column 3: A065091 (odd primes), Column 4: A001248 (squares of primes), Column 6: A006094 (products of two successive primes), Column 8: A030078 (cubes of primes).

Permutations whose formulas refer to this array: A122111, A241909, A242415, A242419, A246676, A246678, A246684.

KEYWORD

nonn,tabl

AUTHOR

Antti Karttunen, May 12 2014

STATUS

approved

A242424

Bulgarian solitaire operation on partition list A112798: a(1) = 1, a(n) = A000040(A001222(n)) * A064989(n).

+10
17

1, 2, 4, 3, 6, 6, 10, 5, 12, 9, 14, 10, 22, 15, 18, 7, 26, 20, 34, 15, 30, 21, 38, 14, 27, 33, 40, 25, 46, 30, 58, 11, 42, 39, 45, 28, 62, 51, 66, 21, 74, 50, 82, 35, 60, 57, 86, 22, 75, 45, 78, 55, 94, 56, 63, 35, 102, 69, 106, 42, 118, 87, 100, 13, 99, 70, 122, 65

(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 A112798.

Please compare to the definition of A122111, which conjugates the partitions encoded with the same system.

a(n) is even if and only if n is either a prime or a multiple of three.

Conversely, a(n) is odd if and only if n is a nonprime not divisible by three.

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..8192

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

Wikipedia, Bulgarian solitaire

Index entries for sequences computed from indices in prime factorization

FORMULA

a(1) = 1, a(n) = A000040(A001222(n)) * A064989(n) = A105560(n) * A064989(n).

a(n) = A241909(A243051(A241909(n))).

a(n) = A243353(A226062(A243354(n))).

a(A000079(n)) = A000040(n) for all n.

A056239(a(n)) = A056239(n) for all n.

PROG

(Scheme) (define (A242424 n) (if (<= n 1) n (* (A000040 (A001222 n)) (A064989 n))))

CROSSREFS

Row 1 of A243070 (table which gives successive "recursive iterates" of this sequence and converges towards A122111).

Fixed points: A002110 (primorial numbers).

Cf. A000041, A243051, A243072, A243073, A226062, A242422, A242423, A122111, A105560, A000040, A001222, A064989, A056239.

KEYWORD

nonn

AUTHOR

Antti Karttunen, May 13 2014

STATUS

approved

A241917

If n is a prime with index i, p_i, a(n) = i, (with a(1)=0), otherwise difference (i-j) of the indices of the two largest primes p_i, p_j, i >= j in the prime factorization of n: a(n) = A061395(n) - A061395(A052126(n)).

+10
16

0, 1, 2, 0, 3, 1, 4, 0, 0, 2, 5, 1, 6, 3, 1, 0, 7, 0, 8, 2, 2, 4, 9, 1, 0, 5, 0, 3, 10, 1, 11, 0, 3, 6, 1, 0, 12, 7, 4, 2, 13, 2, 14, 4, 1, 8, 15, 1, 0, 0, 5, 5, 16, 0, 2, 3, 6, 9, 17, 1, 18, 10, 2, 0, 3, 3, 19, 6, 7, 1, 20, 0, 21, 11, 0, 7, 1, 4, 22, 2, 0, 12, 23

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

OFFSET

1,3

COMMENTS

Note: the two largest primes in the multiset of prime divisors of n are equal for all numbers that are in A070003, thus, after a(1)=0, A070003 gives the positions of the other zeros in this sequence.

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..10000

Index entries for sequences related to prime indices in the factorization of n

FORMULA

a(n) = A061395(n) - A061395(A052126(n)).

PROG

(Scheme) (define (A241917 n) (- (A061395 n) (A061395 (A052126 n))))

(Haskell)

a241917 n = i - j where

(i:j:_) = map a049084 $ reverse (1 : a027746_row n)

-- Reinhard Zumkeller, May 15 2014

(Python)

from sympy import primefactors, primepi

def a061395(n): return 0 if n==1 else primepi(primefactors(n)[-1])

def a052126(n): return 1 if n==1 else n/primefactors(n)[-1]

def a(n): return 0 if n==1 else a061395(n) - a061395(a052126(n)) # Indranil Ghosh, May 19 2017

(PARI) A241917(n) = if(isprime(n), primepi(n), if(1>=omega(n), 0, my(f=factor(n)); if(f[#f~, 2]>1, 0, primepi(f[#f~, 1])-primepi(f[(#f~)-1, 1])))); \\ Antti Karttunen, Jul 10 2024

CROSSREFS

Cf. A052126, A061395, A070003, A122111, A241909.

Cf. A049084, A027746.

Cf. A241919, A242411, A243055 for other variants.

KEYWORD

nonn

AUTHOR

Antti Karttunen, May 13 2014

STATUS

approved

A241918

Table of partitions where the ordering is based on the modified partial sums of the exponents of primes in the prime factorization of n.

+10
15

0, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 3, 1, 2, 2, 2, 2, 1, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 1, 2, 2, 4, 1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 3, 1, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 4, 1, 1, 2, 2, 2, 2, 2, 2, 2, 1, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5

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

OFFSET

1,5

COMMENTS

a(1) = 0 by convention (stands for an empty partition).

For n >= 2, A203623(n-1)+2 gives the index to the beginning of row n and for n>=1, A203623(n)+1 is the index to the end of row n.

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..10081; rows 1..521 flattened.

Marc LeBrun's original "crazy order" mapping for partitions (Copy of Marc's Jan 11 2006 message in OEIS Wiki)

FORMULA

If A241914(n)=0 and A241914(n+1)=0, a(n) = A067255(n); otherwise, if A241914(n)=0 and A241914(n+1)>0, a(n) = A067255(n)+1; otherwise, if A241914(n)>0 and A241914(n+1)=0, a(n) = a(n-1) + A067255(n) - 1, otherwise, when A241914(n)>0 and A241914(n+1)>0, a(n) = a(n-1) + A067255(n).

EXAMPLE

Table begins:

Row Partition

[ 1] 0; (stands for empty partition)

[ 2] 1; (as 2 = 2^1)

[ 3] 1,1; (as 3 = 2^0 * 3^1)

[ 4] 2; (as 4 = 2^2)

[ 5] 1,1,1; (as 5 = 2^0 * 3^0 * 5^1)

[ 6] 2,2; (as 6 = 2^1 * 3^1)

[ 7] 1,1,1,1; (as 7 = 2^0 * 3^0 * 5^0 * 7^1)

[ 8] 3; (as 8 = 2^3)

[ 9] 1,2; (as 9 = 2^0 * 3^2)

[10] 2,2,2; (as 10 = 2^1 * 3^0 * 5^1)

[11] 1,1,1,1,1;

[12] 3,3;

[13] 1,1,1,1,1,1;

[14] 2,2,2,2;

[15] 1,2,2; (as 15 = 2^0 * 3^1 * 5^1)

[16] 4;

[17] 1,1,1,1,1,1,1;

[18] 2,3; (as 18 = 2^1 * 3^2)

etc.

If n is 2^k (k>=1), then the partition is a singleton {k}, otherwise, add one to the exponent of 2 (= A007814(n)), and subtract one from the exponent of the greatest prime dividing n (= A071178(n)), leaving the intermediate exponents as they are, and then take partial sums of all, thus resulting for e.g. 15 = 2^0 * 3^1 * 5^1 the modified sequence of exponents {0+1, 1, 1-1} -> {1,1,0}, whose partial sums {1,1+1,1+1+0} -> {1,2,2} give the corresponding partition at row 15.

MATHEMATICA

Table[If[n == 1, {0}, Function[s, Function[t, Accumulate[If[Length@ t < 2, {0}, Join[{1}, ConstantArray[0, Length@ t - 2], {-1}]] + ReplacePart[t, Map[#1 -> #2 & @@ # &, s]]]]@ ConstantArray[0, Transpose[s][[1, -1]]]][FactorInteger[n] /. {p_, e_} /; p > 0 :> {PrimePi@ p, e}]], {n, 31}] // Flatten (* Michael De Vlieger, May 12 2017 *)

PROG

(Scheme, with Antti Karttunen's IntSeq-library)

(definec (A241918 n) (cond ((zero? (A241914 n)) (if (zero? (A241914 (+ n 1))) (A067255 n) (+ 1 (A067255 n)))) ((zero? (A241914 (+ 1 n))) (+ (A241918 (- n 1)) (- (A067255 n) 1))) (else (+ (A241918 (- n 1)) (A067255 n)))))

CROSSREFS

For n>=2, the length of row n is given by A061395(n).

Cf. also A067255, A203623, A241914.

Other tables of partitions: A112798 (also based on prime factorization), A227739, A242628 (encoded in the binary representation of n), and A036036-A036037, A080576-A080577, A193073 for various lexicographical orderings.

Permutation A241909 maps between order of partitions employed here, and the order employed in A112798.

Permutation A122111 is induced when partitions in this list are conjugated.

A241912 gives the row numbers for which the corresponding rows in A112798 and here are the conjugate partitions of each other.

KEYWORD

nonn,tabf

AUTHOR

Antti Karttunen, May 03 2014, based on Marc LeBrun's Jan 11 2006 message on SeqFan mailing list.

STATUS

approved

A243353

Permutation of natural numbers which maps between the partitions as encoded in A227739 (binary based system, zero-based) to A112798 (prime-index based system, one-based).

+10
15

1, 2, 4, 3, 9, 8, 6, 5, 25, 18, 16, 27, 15, 12, 10, 7, 49, 50, 36, 75, 81, 32, 54, 125, 35, 30, 24, 45, 21, 20, 14, 11, 121, 98, 100, 147, 225, 72, 150, 245, 625, 162, 64, 243, 375, 108, 250, 343, 77, 70, 60, 105, 135, 48, 90, 175, 55, 42, 40, 63, 33, 28, 22, 13, 169, 242, 196, 363, 441, 200, 294, 605, 1225, 450, 144

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

OFFSET

0,2

COMMENTS

Note the indexing: the domain includes zero, but the range starts from one.

LINKS

Antti Karttunen, Table of n, a(n) for n = 0..8192

Index entries for sequences that are permutations of the natural numbers

FORMULA

a(n) = A005940(1+A003188(n)).

a(n) = A241909(1+A075157(n)). [With A075157's original starting offset]

For all n >= 0, A243354(a(n)) = n.

A227183(n) = A056239(a(n)). [Maps between the corresponding sums ...]

A227184(n) = A003963(a(n)). [... and products of parts of each partition].

For n >= 0, a(A037481(n)) = A002110(n). [Also "triangular partitions", the fixed points of Bulgarian solitaire, A226062 & A242424].

For n >= 1, a(A227451(n+1)) = 4*A243054(n).

MATHEMATICA

f[n_, i_, x_] := Which[n == 0, x, EvenQ@ n, f[n/2, i + 1, x], True, f[(n - 1)/2, i, x Prime@ i]]; Table[f[BitXor[n, Floor[n/2]], 1, 1], {n, 0, 74}] (* Michael De Vlieger, May 09 2017 *)

PROG

(Scheme) (define (A243353 n) (A005940 (+ 1 (A003188 n))))

(Python)

from sympy import prime

import math

def A(n): return n - 2**int(math.floor(math.log(n, 2)))

def b(n): return n + 1 if n<2 else prime(1 + (len(bin(n)[2:]) - bin(n)[2:].count("1"))) * b(A(n))

def a005940(n): return b(n - 1)

def a003188(n): return n^int(n/2)

def a243353(n): return a005940(1 + a003188(n)) # Indranil Ghosh, May 07 2017

CROSSREFS

A243354 gives the inverse mapping.

Cf. A227739, A112798, A075157, A241909, A005940, A003188, A226062, A242424.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Jun 05 2014

STATUS

approved

A243503

Sums of parts of partitions (i.e., their sizes) as ordered in the table A241918: a(n) = Sum_{i=A203623(n-1)+2..A203623(n)+1} A241918(i).

+10
15

0, 1, 2, 2, 3, 4, 4, 3, 3, 6, 5, 6, 6, 8, 5, 4, 7, 5, 8, 9, 7, 10, 9, 8, 4, 12, 4, 12, 10, 8, 11, 5, 9, 14, 6, 7, 12, 16, 11, 12, 13, 11, 14, 15, 7, 18, 15, 10, 5, 7, 13, 18, 16, 6, 8, 16, 15, 20, 17, 11, 18, 22, 10, 6, 10, 14, 19, 21, 17, 10, 20, 9, 21, 24, 6, 24

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

OFFSET

1,3

COMMENTS

Each n occurs A000041(n) times in total.

Where are the first and the last occurrence of each n located?

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..8192

FORMULA

a(n) = Sum_{i=A203623(n-1)+2..A203623(n)+1} A241918(i).

a(n) = A056239(A241909(n)).

a(n) = A227183(A075158(n-1)).

a(A000040(n)) = a(A000079(n)) = n for all n >= 1.

a(A122111(n)) = a(n) for all n.

a(A243051(n)) = a(n) for all n, and likewise for A243052, A243053 and other rows of A243060.

a(n) = A061395(n) * A001222(n) + A061395(n) - A056239(n) + A001222(n) - 1. - Gus Wiseman, Jan 09 2023

a(n) = A326844(2n) + A001222(n). - Gus Wiseman, Jan 09 2023

MATHEMATICA

Table[If[n==1, 0, With[{y=Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]}, Last[y]*Length[y]+Last[y]-Total[y]+Length[y]-1]], {n, 100}] (* Gus Wiseman, Jan 09 2023 *)

CROSSREFS

Cf. A243504 (the products of parts), A241918, A000041, A227183, A075158, A056239, A241909.

Sum of prime indices of A241916, the even bisection of A358195.

Sums of even-indexed rows of A358172.

A112798 lists prime indices, length A001222, sum A056239, max A061395.

Cf. A005940, A019565, A246277, A326844, A356958, A359362.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Jun 05 2014

STATUS

approved

A243070

Square array read by antidiagonals: rows are successively recursivized versions of Bulgarian solitaire operation (starting from the usual "first order" version, A242424), as applied to the partitions listed in A112798.

+10
10

1, 2, 1, 4, 2, 1, 3, 4, 2, 1, 6, 3, 4, 2, 1, 6, 8, 3, 4, 2, 1, 10, 6, 8, 3, 4, 2, 1, 5, 12, 6, 8, 3, 4, 2, 1, 12, 5, 16, 6, 8, 3, 4, 2, 1, 9, 9, 5, 16, 6, 8, 3, 4, 2, 1, 14, 12, 9, 5, 16, 6, 8, 3, 4, 2, 1, 10, 20, 12, 9, 5, 16, 6, 8, 3, 4, 2, 1, 22, 10, 24, 12, 9, 5, 16, 6, 8, 3, 4, 2, 1, 15, 28, 10, 32, 12, 9, 5, 16, 6, 8, 3, 4, 2, 1, 18, 18, 40, 10, 32, 12, 9, 5, 16, 6, 8, 3, 4, 2, 1

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

OFFSET

1,2

COMMENTS

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

Please see comments and references in A242424 for more information about Bulgarian Solitaire.

Each row is a A241909-conjugate of the corresponding row in A243060.

Rows in both arrays converge towards A122111.

All the terms in column n are multiples of A105560(n).

The rows of this table (i.e., the corresponding functions) preserve A056239.

First point where row k differs from row k of A243060 seems to be A000040(k+2): primes from five onward: 5, 7, 11, 13, 17, 19, 23, 29, 31, ... and these seem to be also the points where that row differs for the first time from A122111.

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..10440; the first 144 antidiagonals of the square array

FORMULA

A(1,col) = A242424(col), otherwise, when row > 1, A(row,col) = A000040(A001222(col)) * A(row-1, A064989(col)).

EXAMPLE

The top left corner of the array is:

1, 2, 4, 3, 6, 6, 10, 5, 12, 9, 14, 10, 22, 15, 18, ...

1, 2, 4, 3, 8, 6, 12, 5, 9, 12, 20, 10, 28, 18, 18, ...

1, 2, 4, 3, 8, 6, 16, 5, 9, 12, 24, 10, 40, 24, 18, ...

1, 2, 4, 3, 8, 6, 16, 5, 9, 12, 32, 10, 48, 24, 18, ...

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

PROG

(Scheme)

(define (A243070 n) (A243070bi (A002260 n) (A004736 n)))

(define (A243070bi row col) (cond ((<= col 1) col) ((= 1 row) (A242424 col)) (else (* (A000040 (A001222 col)) (A243070bi (- row 1) (A064989 col))))))

CROSSREFS

Row 1: A242424, Row 2: A243072, Row 3: A243073.

Rows converge towards A122111.

Cf. A243060, A241909, A112798, A105560, A000040, A001222, A064989.

KEYWORD

nonn,tabl

AUTHOR

Antti Karttunen, May 29 2014

STATUS

approved

A243354

Permutation of natural numbers which maps between the partitions as encoded in A112798 (prime-index based system, one-based) to A227739 (binary based system, zero-based).

+10
10

0, 1, 3, 2, 7, 6, 15, 5, 4, 14, 31, 13, 63, 30, 12, 10, 127, 9, 255, 29, 28, 62, 511, 26, 8, 126, 11, 61, 1023, 25, 2047, 21, 60, 254, 24, 18, 4095, 510, 124, 58, 8191, 57, 16383, 125, 27, 1022, 32767, 53, 16, 17, 252, 253, 65535, 22, 56, 122, 508, 2046, 131071

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

OFFSET

1,3

COMMENTS

Note the indexing: the domain starts from one, but the range also includes zero.

LINKS

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

Index entries for sequences that are permutations of the natural numbers

FORMULA

a(n) = A006068(A156552(n)).

a(n) = A075158(A241909(n)-1). [With A075158's original starting offset].

For all n >= 1, A243353(a(n)) = n.

A056239(n) = A227183(a(n)).

A003963(n) = A227184(a(n)).

A037481(n) = a(A002110(n)).

PROG

(Scheme) (define (A243354 n) (A006068 (A156552 n)))

CROSSREFS

Inverse: A243353.

Cf. A227739, A112798, A006068, A156552, A075158, A241909.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Jun 05 2014

STATUS

approved

A246684

"Caves of prime shift" permutation: a(1) = 1, a(n) = A242378(A007814(n), 2*a(A003602(n))) - 1.

+10
9

1, 2, 3, 4, 5, 8, 7, 6, 9, 14, 15, 24, 13, 26, 11, 10, 17, 20, 27, 34, 29, 80, 47, 48, 25, 32, 51, 124, 21, 44, 19, 12, 33, 74, 39, 54, 53, 98, 67, 76, 57, 104, 159, 624, 93, 404, 95, 120, 49, 50, 63, 64, 101, 152, 247, 342, 41, 38, 87, 174, 37, 62, 23, 16, 65, 56, 147, 244, 77, 188, 107, 90, 105, 374, 195, 324, 133, 170, 151, 142, 113, 92

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

OFFSET

1,2

COMMENTS

See the comments in A246676. This is otherwise similar permutation, except that after having reached an odd number 2m-1 when we have shifted the binary representation of n right k steps, here, in contrary to A246676, we don't shift the primes in the prime factorization of the even number 2m, but instead of an even number (2*a(m)), shifting it the same number (k) of positions towards larger primes, whose product is then decremented by one to get the final result.

From Antti Karttunen, Jan 18 2015: (Start)

This can be viewed as an entanglement or encoding permutation where the complementary pairs of sequences to be interwoven together are even and odd numbers (A005843/A005408) which are entangled with another complementary pair: even numbers in the order they appear in A253885 and odd numbers in their usual order: (A253885/A005408).

From the above follows also that this sequence can be represented as a binary tree. Each child to the left is obtained by doubling the parent and subtracting one, and each child to the right is obtained by applying A253885 to the parent:

...................2...................

3 4

5......../ \........8 7......../ \........6

/ \ / \ / \ / \

9 14 15 24 13 26 11 10

17 20 27 34 29 80 47 48 25 32 51 124 21 44 19 12

(End)

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..8192

Index entries for sequences that are permutations of the natural numbers

FORMULA

a(1) = 1, a(n) = A242378(A007814(n), 2*a(A003602(n))) - 1. [Where the bivariate function A242378(k,n) changes each prime p(i) in the prime factorization of n to p(i+k), i.e., it's the result of A003961 iterated k times starting from n].

a(1) = 1, a(2n) = A253885(a(n)), a(2n+1) = (2*a(n+1))-1. - Antti Karttunen, Jan 18 2015

As a composition of other permutations:

a(n) = A250243(A249814(n)).

Other identities. For all n >= 1, the following holds:

a(n) = (1+a((2*n)-1))/2. [The odd bisection from a(1) onward with one added and then halved gives the sequence back].

For all n >= 0, the following holds: a(A000051(n)) = A000051(n). [Numbers of the form 2^n + 1 are among the fixed points].

EXAMPLE

Consider n=30, "11110" in binary. It has to be shifted just one bit-position right that the result were an odd number 15, "1111" in binary. As 15 = 2*8-1, we use 2*a(8) = 2*6 = 12 = 2*2*3 = p_1 * p_1 * p_2 [where p_k denotes the k-th prime, A000040(k)], which we shift one step towards larger primes resulting p_2 * p_2 * p_3 = 3*3*5 = 45, thus a(30) = 45-1 = 44.

PROG

(PARI)

A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ Using code of Michel Marcus

A246684(n) = { my(k=0); if(1==n, 1, while(!(n%2), n = n/2; k++); n = 2*A246684((n+1)/2); while(k>0, n = A003961(n); k--); n-1); };

for(n=1, 8192, write("b246684.txt", n, " ", A246684(n)));

(Scheme, with memoization-macro definec, two implementations)

(definec (A246684 n) (cond ((<= n 1) n) (else (+ -1 (A242378bi (A007814 n) (* 2 (A246684 (A003602 n)))))))) ;; Code for A242378bi given in A242378.

(definec (A246684 n) (cond ((<= n 1) n) ((even? n) (A253885 (A246684 (/ n 2)))) (else (+ -1 (* 2 (A246684 (/ (+ n 1) 2)))))))

CROSSREFS

Inverse: A246683.

Other versions: A246676, A246678.

Similar or related permutations: A005940, A163511, A241909, A245606, A246278, A246375, A249814, A250243.

Cf. A000040, A000051, A003602, A003961, A007814, A242378, A253885.

Differs from A249814 for the first time at n=14, where a(14) = 26, while A249814(14) = 20.

KEYWORD

nonn,tabf,look

AUTHOR

Antti Karttunen, Sep 06 2014

STATUS

approved

A165199

a(n) is obtained by flipping every second bit in the binary representation of n starting at the second-most significant bit and on downwards.

+10
8

0, 1, 3, 2, 6, 7, 4, 5, 13, 12, 15, 14, 9, 8, 11, 10, 26, 27, 24, 25, 30, 31, 28, 29, 18, 19, 16, 17, 22, 23, 20, 21, 53, 52, 55, 54, 49, 48, 51, 50, 61, 60, 63, 62, 57, 56, 59, 58, 37, 36, 39, 38, 33, 32, 35, 34, 45, 44, 47, 46, 41, 40, 43, 42, 106, 107, 104, 105, 110, 111, 108

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

OFFSET

0,3

COMMENTS

This is a self-inverse permutation of the positive integers.

Old name was: a(0) = 0, and for n>=1, let b(n,m) be the m-th digit, reading left to right, of binary n. (b(n, 1) is the most significant binary digit, which is 1.) Then a(n) is such that b(a(n),1)=1; and if b(n,m)=b(n,m-1) then b(a(n),m) does not = b(a(n),m-1); and if b(n,m) does not = b(n,m-1) then b(a(n), m) = b(a(n),m-1), for all m where 2 <= m <= number binary digits in n.

From Emeric Deutsch, Oct 06 2020: (Start)

a(n) is the index of the composition that is the conjugate of the composition with index n.

The index of a composition is defined to be the positive integer whose binary form has run-lengths (i.e., runs of 1's, runs of 0's, etc. from left to right) equal to the parts of the composition. Example: the composition 1,1,3,1 has index 46 since the binary form of 46 is 101110.

a(18) = 24. Indeed, since the binary form of 18 is 10010, the composition with index 18 is 1,2,1,1 (the run-lengths of 10010); the conjugate of 1,2,1,1 is 2,3 and so the binary form of a(18) is 11000; consequently, a(18) = 24. (End)

LINKS

Antti Karttunen, Table of n, a(n) for n = 0..1023

Index entries for sequences related to binary expansion of n

Index entries for sequences that are permutations of the natural numbers

FORMULA

From Antti Karttunen, Jul 22 2014: (Start)

a(0) = 0, and for n >= 1, a(n) = 2*a(floor(n/2)) + A000035(n+A000523(n)).

As a composition of related permutations:

a(n) = A056539(A129594(n)) = A129594(A056539(n)).

a(n) = A245443(A193231(n)) = A193231(A245444(n)).

a(n) = A075158(A243353(n)-1) = A075158((A241909(1+A075157(n))) - 1).

(End)