A059893 -id:A059893 - cp's OEIS frontend

A065275 Infinite binary tree inspired permutation of N: 1 -> 3, 11ab..yz -> 11ab..yz0, 10ab..y0 -> 10ab..y, 10ab..y1 -> 11ab..y1.

3, 1, 6, 2, 7, 12, 14, 4, 13, 5, 15, 24, 26, 28, 30, 8, 25, 9, 27, 10, 29, 11, 31, 48, 50, 52, 54, 56, 58, 60, 62, 16, 49, 17, 51, 18, 53, 19, 55, 20, 57, 21, 59, 22, 61, 23, 63, 96, 98, 100, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120, 122, 124, 126, 32, 97, 33, 99

Offset: 1

Antti Karttunen, Oct 28 2001

nonn

On the right side every node replaces its left child, on the left side the left children replace their parents and the right children are transferred to the same offset at the right side (staying right children). See comment at A065263.

Index entries for sequences that are permutations of the natural numbers

A057114, A065263, A065269, A065281, A065287. Inverse: A065276, conjugated with A059893: A065277 and the inverse of that: A065278.

RightChildTransferred := proc(n) local k; if(1 = n) then RETURN(3); fi; k := floor_log_2(n)-1; if(3 = floor(n/(2^k))) then RETURN(2*n); fi; if(0 = (n mod 2)) then RETURN(n/2); fi; RETURN(n + (2^k)); end;

A065281 Infinite binary tree inspired permutation of N: 1 -> 1, 11ab..yz -> 11ab..yz1, 10ab..y1 -> 10ab..y, 10ab..y0 -> 11ab..y0.

Original entry on oeis.org

1, 3, 7, 6, 2, 13, 15, 12, 4, 14, 5, 25, 27, 29, 31, 24, 8, 26, 9, 28, 10, 30, 11, 49, 51, 53, 55, 57, 59, 61, 63, 48, 16, 50, 17, 52, 18, 54, 19, 56, 20, 58, 21, 60, 22, 62, 23, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125, 127, 96, 32, 98, 33

Offset: 1

Antti Karttunen, Oct 28 2001

nonn

On the right side every node replaces its right child, on the left side the right children replace their parents and the left children are transferred to the same offset at the right side (staying left children). See comment at A065263.

Index entries for sequences that are permutations of the natural numbers

A057114, A065263, A065269, A065275, A065287. Inverse: A065282, conjugated with A059893: A065283 and the inverse of that: A065284.

LeftChildTransferred := proc(n) local k; if(1 = n) then RETURN(1); fi; k := floor_log_2(n)-1; if(3 = floor(n/(2^k))) then RETURN((2*n)+1); fi; if(1 = (n mod 2)) then RETURN((n-1)/2); fi; RETURN(n + (2^k)); end;

A065287 Infinite binary tree inspired permutation of N: 1 -> 3, 11ab..yz -> 11ab..yz1, 10ab..y0 -> 10ab..y, 10ab..y1 -> 11ab..y0.

Original entry on oeis.org

3, 1, 7, 2, 6, 13, 15, 4, 12, 5, 14, 25, 27, 29, 31, 8, 24, 9, 26, 10, 28, 11, 30, 49, 51, 53, 55, 57, 59, 61, 63, 16, 48, 17, 50, 18, 52, 19, 54, 20, 56, 21, 58, 22, 60, 23, 62, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125, 127, 32, 96, 33, 98

Offset: 1

Antti Karttunen, Oct 28 2001

nonn

On the right side every node replaces its right child, on the left side the left children replace their parents and the right children are transferred to the same offset - 1 at the right side (becoming left children). See comment at A065263.

Index entries for sequences that are permutations of the natural numbers

A057114, A065263, A065269, A065275, A065281. Inverse: A065288, conjugated with A059893: A065289 and the inverse of that: A065290.

RightChildTransferred_1 := proc(n) local k; if(1 = n) then RETURN(3); fi; k := floor_log_2(n)-1; if(3 = floor(n/(2^k))) then RETURN((2*n)+1); fi; if(0 = (n mod 2)) then RETURN(n/2); fi; RETURN(n + (2^k) - 1); end;

A343152 Reverse the order of all but the most significant bits in the maximal Fibonacci expansion of n.

Original entry on oeis.org

1, 2, 3, 4, 6, 5, 7, 8, 11, 10, 9, 12, 16, 14, 19, 13, 18, 17, 15, 20, 21, 29, 27, 24, 32, 26, 23, 31, 22, 30, 28, 25, 33, 42, 37, 50, 35, 48, 45, 40, 53, 34, 47, 44, 39, 52, 43, 38, 51, 36, 49, 46, 41, 54, 55, 76, 71, 63, 84, 69, 61, 82, 58, 79, 74, 66, 87

Offset: 1

J. Parker Shectman, Apr 07 2021

A self-inverse permutation of the natural numbers.
Analogous to A059893 with binary expansion replaced by maximal Fibonacci expansion.
Analogous to A343150 with minimal Fibonacci expansion replaced by maximal Fibonacci expansion.
For n=1, the expansion equals 1. For n>=2, the expansion equals A104326(n-1) with a 1 appended. The 1 corresponds to a digit (always equal to 1) for F(1)=1, in addition to the digit for F(2)=1. (This expansion is NOT a representation, see reference in link, pp. 106 and 137.)
Write the sequence as a (right-justified) "tetrangle" or "irregular triangle" tableau with F(t) (Fibonacci number) entries on each row, for t=1,2,3,.... Then, columns of the tableau equal rows of the array A083047 (see reference in link, p. 131):
                               1
                               2
                           3,  4
                       6,  5,  7
               8, 11, 10,  9, 12
  16, 14, 19, 13, 18, 17, 15, 20
  ...

			For an example of calculation by reversing Fibonacci binary digits, see reference in link, p. 144:
On the basis (1,1,2,3,5,8) n=13 is written as 110101, Reversing all but the most AND least significant digits gives 101011, which evaluates to 16, so a(13)=16.
On the basis (1,1,2,3,5,8) n=14 is written as 101101, Reversing all but the most AND least significant digits gives 101101, which evaluates to 14, so a(14)=14.

Cf. A059893, A083047, A104326, A247647, A343150.

(*Produce indices of maximal Fibonacci expansion (recursively)*)
MaxFibInd[n_] := Module[{t = Floor[Log[GoldenRatio, Sqrt[5]*n + 1]] - 1}, Piecewise[{{{1}, n == 1}, {Append[MaxFibInd[n - Fibonacci[t]], t], n > 1}},]];
(*Define a(n)*)
a[n_] := Module[{MFI = MaxFibInd[n]}, Apply[Plus, Fibonacci[Last[MFI] - MFI + 1]]];
(*Generate DATA*)
Array[a, 67]

A358170 Heinz number of the partial sums of the n-th composition in standard order (A066099).

Original entry on oeis.org

1, 2, 3, 6, 5, 15, 10, 30, 7, 35, 21, 105, 14, 70, 42, 210, 11, 77, 55, 385, 33, 231, 165, 1155, 22, 154, 110, 770, 66, 462, 330, 2310, 13, 143, 91, 1001, 65, 715, 455, 5005, 39, 429, 273, 3003, 195, 2145, 1365, 15015, 26, 286, 182, 2002, 130, 1430, 910, 10010

Offset: 0

Gus Wiseman, Dec 20 2022

nonn

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

			The terms together with their prime indices begin:
           1: {}
           2: {1}
           3: {2}
           6: {1,2}
           5: {3}
          15: {2,3}
          10: {1,3}
          30: {1,2,3}
           7: {4}
          35: {3,4}
          21: {2,4}
         105: {2,3,4}
          14: {1,4}
          70: {1,3,4}
          42: {1,2,4}
         210: {1,2,3,4}

Gus Wiseman, Statistics, classes, and transformations of standard compositions

See link for sequences related to standard compositions.
Applying A001221 or A001222 gives A000120.
The image is A005117 (squarefree numbers).
The reverse version is A019565, triangular version A048793.
Greatest prime index of a(n) is A029837 or A070939.
Least prime index of a(n) is A065120.
The adjusted version is A253565, inverse A253566, reverse A005940.
These are the Heinz numbers of the rows of A358134.
Sum of prime indices of a(n) is A359042.
A066099 lists standard compositions.
A112798 list prime indices, sum A056239.
Cf. A000720, A001511, A029931, A059893, A061395, A241916, A242628, A355536, A358133, A358135, A358137.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Times@@Prime/@#&/@Table[Accumulate[stc[n]],{n,0,100}]

A153152 Rotated binary incrementing: For n<2 a(n)=n, if n=(2^k)-1, a(n)=(n+1)/2, otherwise a(n)=n+1.

Original entry on oeis.org

0, 1, 3, 2, 5, 6, 7, 4, 9, 10, 11, 12, 13, 14, 15, 8, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 16, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 32, 65, 66, 67, 68, 69, 70, 71, 72

Offset: 0

Antti Karttunen, Dec 20 2008

A002487(n)/A002487(n+1), n > 0, runs through all the reduced nonnegative rationals exactly once. A002487 is the Stern's sequence. A002487(a(n)) = A002487(n+1) n>0 . - Yosu Yurramendi, Jul 07 2016

Inverse: A153151.

a(n) = A059893(A153142(A059893(n))) = A059894(A153141(A059894(n))).

a := n -> if n < 2 then n elif convert(convert(n+1, base, 2), `+`) = 1 then (n+1)/2 else n+1 fi: seq(a(n), n=0..71); # Peter Luschny, Jul 16 2016

Table[If[IntegerQ@ Log2[n + 1], (n + 1)/2, n + 1], {n, 0, 71}] /. Rational -> 0 (* _Michael De Vlieger, Jul 13 2016 *)

def ok(n): return n&(n - 1)==0
def a(n): return n if n<2 else (n + 1)/2 if ok(n + 1) else n + 1 # Indranil Ghosh, Jun 09 2017

maxlevel <- 5 # by choice
a <- 1
for(m in 1:maxlevel){
 a[2^m        ] <- 2^m + 1
 a[2^(m+1) - 1] <- 2^m
 for (k in 0:(2^m-2)){
   a[2^(m+1) + 2*k + 1] <- 2*a[2^m + k]
   a[2^(m+1) + 2*k + 2] <- 2*a[2^m + k] + 1}
}
a <- c(0, a)
# Yosu Yurramendi, Sep 05 2020

A258015 Capped binary boundary codes for those fusenes that stay same when flipped over, only the maximal representative from each equivalence class up to rotation.

Original entry on oeis.org

1, 127, 2014, 7918, 31606, 32122, 32188, 126394, 486838, 503482, 505564, 506332, 511708, 511804, 513514, 514936, 2012890, 2021098, 2025196, 2054044, 2055544, 7788250, 8050522, 8051434, 8051548, 8054620, 8075098, 8075110, 8084380, 8104888, 8182636, 8183020, 8185756, 8207218, 8207602, 8214442, 8219596, 8219602, 8231884, 8236516, 8238832

Offset: 0

Antti Karttunen, Jun 01 2015

These are binary boundary codes for fusenes with bilateral symmetry, i.e., those terms k in A258013 for which A256999(A059893(k)) = k. A258018(n) gives the count of terms with binary width 2n + 1.

Differs from its subsequence A258005 for the first time at n=113, as a(113) = 131821024 is the first term not present in A258005.

Antti Karttunen, Table of n, a(n) for n = 0..113

Subsequence of A258013.
Subsequence: A258005.
Cf. A258018.

A334032 The a(n)-th composition in standard order (graded reverse-lexicographic) is the unsorted prime signature of n.

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 1, 4, 2, 3, 1, 5, 1, 3, 3, 8, 1, 6, 1, 5, 3, 3, 1, 9, 2, 3, 4, 5, 1, 7, 1, 16, 3, 3, 3, 10, 1, 3, 3, 9, 1, 7, 1, 5, 5, 3, 1, 17, 2, 6, 3, 5, 1, 12, 3, 9, 3, 3, 1, 11, 1, 3, 5, 32, 3, 7, 1, 5, 3, 7, 1, 18, 1, 3, 6, 5, 3, 7, 1, 17, 8, 3, 1, 11

Offset: 1

Gus Wiseman, Apr 17 2020

nonn

Unsorted prime signature (A124010) is the sequence of exponents in a number's prime factorization.

The k-th composition in standard order (row k of A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			The unsorted prime signature of 12345678 is (1,2,1,1), which is the 27th composition in standard order, so a(12345678) = 27.

Positions of first appearances are A057335 (a partial inverse).
Least number with same prime signature is A071364.
Unsorted prime signature is A124010.
Least number with reversed prime signature is A331580.
Minimal numbers with standard reversed prime signatures are A334031.
The reversed version is A334033.
All of the following pertain to compositions in standard order (A066099):
- Length is A000120.
- Sum is A070939.
- Strict compositions are A233564.
- Constant compositions are A272919.
- Aperiodic compositions are A328594.
- Normal compositions are A333217.
- Permutations are A333218.
- Heinz number is A333219.
Cf. A029931, A048793, A052409, A055932, A056239, A112798, A124767, A228351, A233249, A329139, A333220.

stcinv[q_]:=Total[2^Accumulate[Reverse[q]]]/2;
Table[stcinv[Last/@If[n==1,{},FactorInteger[n]]],{n,100}]

a(A057335(n)) = n.
A057335(a(n)) = A071364(n).
a(A334031(n))= A059893(n).
A334031(a(n)) = A331580(n).

A258005 Capped binary boundary codes for holeless strictly non-overlapping polyhexes with bilateral symmetry, only the maximal representative from each equivalence class obtained by rotating.

Original entry on oeis.org

1, 127, 2014, 7918, 31606, 32122, 32188, 126394, 486838, 503482, 505564, 506332, 511708, 511804, 513514, 514936, 2012890, 2021098, 2025196, 2054044, 2055544, 7788250, 8050522, 8051434, 8051548, 8054620, 8075098, 8075110, 8084380, 8104888, 8182636, 8183020, 8185756, 8207218, 8207602, 8214442, 8219596, 8219602, 8231884, 8236516

Offset: 0

Antti Karttunen, May 31 2015

Indexing starts from zero, because a(0) = 1 is a special case, indicating an empty path in the honeycomb lattice.
These are capped binary boundary codes for those holeless polyhexes that stay same when they are flipped over and rotated appropriately.
A258205(n) gives the count of terms with binary width 2n + 1.

Antti Karttunen, Table of n, a(n) for n = 0..112

Cf. A255571, A258205.
Intersection of A258003 and A258209. Differs from A258003 for the first time at n=8, where a(8) = 486838 while A258003(8) = 127930.
Subsequence of A258015 from which this differs for the first time at n=113.

A273493 a(n) = A245327(n) + A245328(n).

Original entry on oeis.org

2, 3, 3, 5, 5, 4, 4, 8, 8, 7, 7, 7, 7, 5, 5, 13, 13, 11, 11, 12, 12, 9, 9, 11, 11, 10, 10, 9, 9, 6, 6, 21, 21, 18, 18, 19, 19, 14, 14, 19, 19, 17, 17, 16, 16, 11, 11, 18, 18, 15, 15, 17, 17, 13, 13, 14, 14, 13, 13, 11, 11, 7, 7, 34, 34, 29, 29, 31, 31, 23, 23, 30, 30, 27, 27, 25, 25, 17, 17, 31, 31, 26, 26, 29, 29, 22, 22, 25

Offset: 1

Yosu Yurramendi, May 23 2016

nonn

The terms (n>0) may be written as a left-justified array with rows of length 2^m, m >= 0:
2,
3, 3,
5, 5, 4, 4,
8, 8, 7, 7, 7, 7, 5, 5,
13,13,11,11,12,12, 9, 9,11,11,10,10, 9, 9, 6, 6,
21,21,18,18,19,19,14,14,19,19,17,17,16,16,11,11,18,18,15,15,17,17,13,13,14,14,...
All columns have the Fibonacci sequence property: a(2^(m+2) + k) = a(2^(m+1) + k) + a(2^m + k), m >= 0,  0 <= k < 2^m (empirical observations).
The terms (n>0) may also be written as a right-justified array with rows of length 2^m, m >= 0:
                                                                              2,
                                                                           3, 3,
                                                                     5, 5, 4, 4,
                                                         8, 8, 7, 7, 7, 7, 5, 5,
                                13,13,11,11,12,12, 9, 9,11,11,10,10, 9, 9, 6, 6,
    ...,19,19,17,17,16,16,11,11,18,18,15,15,17,17,13,13,14,14,13,13,11,11, 7, 7,
Each column is an arithmetic sequence. The differences of the arithmetic sequences repeat the sequence A071585: a(2^(m+2) -1 - 2k) - a(2^(m+1) -1 - 2k) = A071585(k-1),    m > 0,  0 <= k < 2^m ; a(2^(m+2) -1 - 2k - 1) - a(2^(m+1) -1 - 2k - 1) = A071585(k-1), m > 0,  0 <= k < 2^m .
n>1 occurs in this sequence phi(n) = A000010(n) times, as it occurs in A007306 (Franklin T. Adams-Watters' comment), that is the sequence obtained by adding numerator and denominator in the Calkin-Wilf enumeration system of positive rationals. A245327(n)/A245328(n) is also an enumeration system of all positive rationals, and in each level m >= 0 (ranks between 2^m and 2^(m+1)-1) rationals are the same in both systems. Thus a(n) has the same terms in each level as A007306.
a(n) = A273494(A059893(n)), a(A059893(n)) = A273494(n), n > 0. - Yosu Yurramendi, May 30 2017

Yosu Yurramendi, Table of n, a(n) for n = 1..4099

Cf. A007306, A268087, A071585, A086592, A273494

b(n) = my(b=binary(n)); fromdigits(concat(b[1], Vecrev(vector(#b-1, k, b[k+1]))), 2); \\ from A059893
a(n) = my(n=b(n), x=1, y=1); for(i=0, logint(n, 2), if(bittest(n, i), [x, y]=[x+y, y], [x, y]=[y, x+y])); x \\ Mikhail Kurkov, Mar 11 2023

a(n) = A007306(A284459(n)), n > 0. - Yosu Yurramendi, Aug 23 2021

cp's OEIS Frontend

A065275 Infinite binary tree inspired permutation of N: 1 -> 3, 11ab..yz -> 11ab..yz0, 10ab..y0 -> 10ab..y, 10ab..y1 -> 11ab..y1.

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A065281 Infinite binary tree inspired permutation of N: 1 -> 1, 11ab..yz -> 11ab..yz1, 10ab..y1 -> 10ab..y, 10ab..y0 -> 11ab..y0.

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A065287 Infinite binary tree inspired permutation of N: 1 -> 3, 11ab..yz -> 11ab..yz1, 10ab..y0 -> 10ab..y, 10ab..y1 -> 11ab..y0.

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A343152 Reverse the order of all but the most significant bits in the maximal Fibonacci expansion of n.

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A358170 Heinz number of the partial sums of the n-th composition in standard order (A066099).

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A153152 Rotated binary incrementing: For n<2 a(n)=n, if n=(2^k)-1, a(n)=(n+1)/2, otherwise a(n)=n+1.

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A258015 Capped binary boundary codes for those fusenes that stay same when flipped over, only the maximal representative from each equivalence class up to rotation.

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A334032 The a(n)-th composition in standard order (graded reverse-lexicographic) is the unsorted prime signature of n.

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A258005 Capped binary boundary codes for holeless strictly non-overlapping polyhexes with bilateral symmetry, only the maximal representative from each equivalence class obtained by rotating.

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