A241909 -id:A241909 - cp's OEIS frontend

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).

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

Offset: 0

Antti Karttunen, May 12 2014

Each row i is a multiplicative function, being in essence "the i-th power" of A003961, i.e., A(i,j) = A003961^i (j). 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), ... .

			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.

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

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).
Excluding column 0, a subtable of A297845.
Permutations whose formulas refer to this array: A122111, A241909, A242415, A242419, A246676, A246678, A246684.

A(0,j) = j, A(i,0) = 0, A(i > 0, j > 0) = A003961(A(i-1,j)).

For j > 0, A(i,j) = A297845(A000040(i+1),j) = A297845(j,A000040(i+1)). - Peter Munn, Sep 02 2025

A242424 Bulgarian solitaire operation on partition list A112798: a(1) = 1, a(n) = A000040(A001222(n)) * A064989(n).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, May 13 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 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.

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..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

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.

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

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.

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)).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, May 13 2014

nonn

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.

Cf. A052126, A061395, A070003, A122111, A241909.
Cf. A049084, A027746.
Cf. A241919, A242411, A243055 for other variants.

a241917 n = i - j where
            (i:j:_) = map a049084 $ reverse (1 : a027746_row n)
-- Reinhard Zumkeller, May 15 2014

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

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

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

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

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.

Original entry on oeis.org

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

Offset: 1

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

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.

			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.

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)

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.

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 *)

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).

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).

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Jun 05 2014

nonn

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

Antti Karttunen, Table of n, a(n) for n = 0..8192
Index entries for sequences that are permutations of the natural numbers

A243354 gives the inverse mapping.

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

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 *)

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

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

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).

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).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jun 05 2014

nonn

Each n occurs A000041(n) times in total.

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

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

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.

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 *)

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

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.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, May 29 2014

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.

			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, ...

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

Row 1: A242424, Row 2: A243072, Row 3: A243073.
Rows converge towards A122111.
Cf. A243060, A241909, A112798, A105560, A000040, A001222, A064989.

(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))))))

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

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).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jun 05 2014

nonn

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

Antti Karttunen, Table of n, a(n) for n = 1..1024
Index entries for sequences that are permutations of the natural numbers

Inverse: A243353.

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

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

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)).

A246684 "Caves of prime shift" permutation: a(1) = 1, a(n) = A242378(A007814(n), 2*a(A003602(n))) - 1.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Sep 06 2014

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:
                                      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
(End)

			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.

Antti Karttunen, Table of n, a(n) for n = 1..8192
Index entries for sequences that are permutations of the natural numbers

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.

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)))))))

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].

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.

Original entry on oeis.org

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

Offset: 0

Leroy Quet, Sep 07 2009

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)

			a(12) = 9 because 12 = 1100_2 and 1100_2 XOR 0101_2 = 1001_2 = 9.

Cf. A000035, A000523, A075157, A075158, A241909, A243353, A245443, A245444.

{A001477, A129594, A165199, A056539} form a 4-group.

a:= n-> Bits[Xor](n, iquo(2^(1+ilog2(n)), 3)):
seq(a(n), n=0..100);  # Alois P. Heinz, Oct 07 2020

for(k=0,67,my(b(n)=vector(#digits(n,2),i,!(i%2)));print1(bitxor(k,fromdigits(b(k),2)),", ")) \\ Hugo Pfoertner, Oct 07 2020

a(n) = if(n, bitxor(n,2<Kevin Ryde, Oct 07 2020

maxrow <- 8 # by choice
a <- 1
for(m in 0: maxrow) for(k in 0:(2^m-1)){
a[2^(m+1) +       k] = a[2^(m+1) - 1 - k] + 2^(m+1)
a[2^(m+1) + 2^m + k] = a[2^(m+1) - 1 - k] + 2^m
}
(a <- c(0, a))
# Yosu Yurramendi, Apr 04 2017

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)
a(n) = A258746(A054429(n)) = A054429(A258746(n)), n > 0. - Yosu Yurramendi, Mar 29 2017

Extended by Ray Chandler, Sep 10 2009
a(0) = 0 prepended by Antti Karttunen, Jul 22 2014
New name from Kevin Ryde, Oct 07 2020

cp's OEIS Frontend

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).

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A242424 Bulgarian solitaire operation on partition list A112798: a(1) = 1, a(n) = A000040(A001222(n)) * A064989(n).

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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)).

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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.

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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).

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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).

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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.

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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).

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