A056539 -id:A056539 - cp's OEIS frontend

A227987 If the run lengths of the binary representation of n are [1+r_1, 1+r_2, 1+r_3, ..., 1+r_k], then those of a(n) are [1+(r_1), 1+(r_1 XOR r_2), 1+(r_1 XOR r_2 XOR r_3), ..., 1+(r_1 XOR ... XOR r_k)], where XOR denotes the XOR binary operator.

1, 2, 3, 4, 5, 12, 7, 8, 19, 10, 11, 6, 51, 56, 15, 16, 71, 76, 9, 20, 21, 44, 23, 48, 13, 204, 25, 112, 455, 240, 31, 32, 271, 568, 143, 38, 307, 18, 79, 40, 83, 42, 43, 22, 179, 184, 47, 24, 783, 26, 27, 102, 819, 50, 207, 14, 1807, 3640, 911, 120, 3855

Offset: 1

Paul Tek, Aug 02 2013

This is a permutation of the natural numbers with inverse permutation A225607.
The sequence (n, a(n), a(a(n)), a(a(a(n))),...) is periodic for any n.
The run lengths of the binary representation of a fixed point are of the form [1, 1,...,1, K] (any number of ones followed by any number).

			For n=927:
(1) binary representation of n = "1110011111",
(2) run lengths of n = [1+2,1+1,1+4],
(3) run lengths of a(n) = [1+(2),1+(2 XOR 1),1+(2 XOR 1 XOR 4)]=[3,4,8],
(4) binary representation of a(n) = "111000011111111",
(5) a(n) = 28927.

Paul Tek, Table of n, a(n) for n = 1..10000
Paul Tek, Perl program for this sequence
Index entries for sequences that are permutations of the natural numbers

Cf. A056539, A226532, A225607 (inverse).

Perl
```
# See Tek link.
```

A322463 Reverse runs of zeros in binary expansion of n and convert back to decimal.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 20, 19, 18, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 40, 35, 36, 41, 44, 39, 34, 37, 42, 43, 38, 45, 46, 47, 48, 49, 52, 51, 50, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 80, 67

Offset: 0

Rémy Sigrist, Dec 09 2018

This sequence is a self-inverse permutation of nonnegative integers.

			For n = 150:
- the binary representation of 150 is "10010110",
- we have three runs of zeros: "00", "0" and "0",
- we exchange the first and the third run, and the second remains in place,
- we obtain: "10101100",
- hence a(150) = 172.

Rémy Sigrist, Table of n, a(n) for n = 0..8192
Rémy Sigrist, Colored scatterplot of (n, a(n)) for n = 0..2^18-1 (where the color is function of A005811(n))
Index entries for sequences related to binary expansion of n
Index entries for sequences that are permutations of the natural numbers

See A322464 for the variant where we reverse the runs of ones.
See A056539 for a similar sequence.
Cf. A000120, A005811, A164710.

a[n_] := Module[{s=Split[IntegerDigits[n,2]]}, m=Length[s]; m2=m-Mod[m,2]; If[m2>0, ind=Riffle[Range[1,m,2],Range[m2,1,-2]]; FromDigits[Flatten[s[[ind]]],2],n]]; Array[a, 100, 0] (* Amiram Eldar, Dec 12 2018 *)

a(n) = {
    my (r=n, z=[], v=0, p=1, i=0);
    while (r, my (l=valuation(r+(r%2),2)); if (r%2==0, z=concat(l,z)); r\=2^l);
    while (n, my (l=valuation(n+(n%2),2)); if (n%2, v+=(2^l-1)*p; p*=2^l, p*=2^z[i++]); n\=2^l);
    return (v);
}

A000120(a(n)) = A000120(n).
A005811(a(n)) = A005811(n).
a(A164710(n)) = A164710(n).
a(A322464(n)) = A322464(a(n)).
a(2^n) = 2^n.
a(2^n-1) = 2^n-1.

A285916 Self-inverse permutation: in base 3, reverse the lengths of runs of consecutive equal digits.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 12, 10, 11, 9, 13, 17, 15, 16, 14, 24, 19, 20, 21, 25, 23, 18, 22, 26, 39, 28, 29, 30, 37, 32, 33, 34, 38, 36, 31, 35, 27, 40, 53, 45, 49, 44, 42, 46, 47, 48, 43, 50, 51, 52, 41, 78, 55, 56, 57, 73, 59, 60, 61, 74, 75, 64, 65, 66, 79

Offset: 0

Rémy Sigrist, Apr 28 2017

This sequence can be seen as an analog of A056539 for the base 3.

For any n>=0, n and a(n) have the same number of digits in base 3.

			The first terms for which n<>a(n) are:
Decimal:    Base 3:
n   a(n)    n       a(n)
-   ----    ---     ----
9   12      100     110
12  9       110     100
14  17      112     122
17  14      122     112
18  24      200     220
22  25      211     221
24  18      220     200
25  22      221     211
27  39      1000    1110
31  37      1011    1101
35  38      1022    1102
37  31      1101    1011
38  35      1102    1022
39  27      1110    1000
41  53      1112    1222
42  45      1120    1200

Rémy Sigrist, Table of n, a(n) for n = 0..6560
Rémy Sigrist, PARI program for A285916
Index entries for sequences that are permutations of the natural numbers

Cf. A056539.

A290078 Where the ratio A235027(n)/n obtains record values.

Original entry on oeis.org

1, 11, 19, 67, 71, 263, 271, 781, 1273, 1349, 2981, 4757, 5041, 18157, 18673, 19241, 55451, 71273, 73441, 95779, 211651, 337747, 357911, 1289147, 1325783, 1366111, 3937021, 5060383, 5214311, 6800309, 15027221, 19314983, 19902511, 23980037, 25411681

Offset: 1

Antti Karttunen, Aug 07 2017

Because A056539(n)/n < 2 for all n, and already for the tenth term of this sequence 1349 we have A235027(1349)/1349 = 2.094... it follows that the only primes present are terms a(2) .. a(7): 11, 19, 67, 71, 263, 271. Conjecture: every term after that is a product of some of those six primes. For example: 781 = 11*71, 1273 = 19*67, 1349 = 19*71, 2981 = 11*271, 4757 = 67*71, 5041 = 71*71.

Index entries for sequences related to binary expansion of n

Cf. A235027.

revbits(n) = fromdigits(Vecrev(binary(n)), 2);
A235027(n) = {my(f = factor(n)); for (k=1, #f~, if (f[k, 1] != 2, f[k, 1] = revbits(f[k, 1]); ); ); factorback(f); } \\ This function from Michel Marcus, Aug 05 2017
m=0; i=0; n=0; while(i<35, n++; if((A235027(n)/n) > m, m = A235027(n)/n; i++; print1(n,",")));

A335858 Nonnegative integers ordered by binary length and then lexicographically by run lengths (considering least significant runs first).

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 4, 7, 10, 13, 9, 14, 11, 12, 8, 15, 21, 26, 18, 29, 22, 25, 17, 30, 20, 27, 19, 28, 23, 24, 16, 31, 42, 53, 37, 58, 45, 50, 34, 61, 41, 54, 38, 57, 46, 49, 33, 62, 43, 52, 36, 59, 44, 51, 35, 60, 40, 55, 39, 56, 47, 48, 32, 63, 85, 106, 74

Offset: 0

Rémy Sigrist, Jun 27 2020

The variant where we consider most significant runs first apparently corresponds to A180200.

			The first terms, alongside the corresponding binary representation and run lengths, are:
  n   a(n)  bin(a(n))  A227736(n, *)
  --  ----  ---------  -------------
   0     0          0  ()
   1     1          1  (1)
   2     2         10  (1, 1)
   3     3         11  (2)
   4     5        101  (1, 1, 1)
   5     6        110  (1, 2)
   6     4        100  (2, 1)
   7     7        111  (3)
   8    10       1010  (1, 1, 1, 1)
   9    13       1101  (1, 1, 2)
  10     9       1001  (1, 2, 1)
  11    14       1110  (1, 3)
  12    11       1011  (2, 1, 1)
  13    12       1100  (2, 2)
  14     8       1000  (3, 1)
  15    15       1111  (4)

Rémy Sigrist, Table of n, a(n) for n = 0..8192
Rémy Sigrist, PARI program for A335858
Index entries for sequences related to binary expansion of n
Index entries for sequences that are permutations of the natural numbers

Cf. A005811, A180200, A227736.

PARI
```
See Links section.
```

Apparently a(n) = A056539(A180200(n)).

A337672 Numbers with binary expansion Sum_{k = 0..w} b_k * 2^k such that the polynomial Sum_{k = 0..w} (X+k)^2 * (-1)^b_k is constant.

Original entry on oeis.org

0, 9, 150, 153, 165, 195, 2268, 2282, 2289, 2364, 2394, 2406, 2409, 2454, 2457, 2469, 2499, 2618, 2646, 2649, 2661, 2702, 2709, 2723, 2829, 2835, 3126, 3129, 3150, 3157, 3171, 3213, 3219, 3339, 3591, 34680, 34740, 34764, 34770, 34785, 35576, 35700, 35756

Offset: 1

Rémy Sigrist, Sep 15 2020

Leading 0's in binary expansions are ignored.
Positive terms are digitally balanced (A031443).
If m belongs to the sequence, then A056539(m) also belongs to the sequence.
If m and n belong to the sequence, then their binary concatenation also belongs to the sequence (assuming the concatenation with 0 is neutral).

			The first 16 integers, alongside their binary representations and associate polynomials, are:
  k   bin(k)  P(k)
  --  ------  --------------
   0       0               0
   1       1            -X^2
   2      10           2*X+1
   3      11    -2*X^2-2*X-1
   4     100       X^2+6*X+5
   5     101      -X^2-2*X-3
   6     110      -X^2+2*X+3
   7     111    -3*X^2-6*X-5
   8    1000   2*X^2+12*X+14
   9    1001              -4
  10    1010           4*X+6
  11    1011   -2*X^2-8*X-12
  12    1100          8*X+12
  13    1101    -2*X^2-4*X-6
  14    1110        -2*X^2+4
  15    1111  -4*X^2-12*X-14
We have constant polynomials for k = 0 and k = 9, so a(1) = 0 and a(2) = 9.

Index entries for sequences related to binary expansion of n

Cf. A031443, A056539, A133468.

is(n) = { my (b=Vecrev(binary(n))); poldegree(p=sum(k=1, #b, ('X+k-1)^2 * (-1)^b[k]))<=0 }

cp's OEIS Frontend

A227987 If the run lengths of the binary representation of n are [1+r_1, 1+r_2, 1+r_3, ..., 1+r_k], then those of a(n) are [1+(r_1), 1+(r_1 XOR r_2), 1+(r_1 XOR r_2 XOR r_3), ..., 1+(r_1 XOR ... XOR r_k)], where XOR denotes the XOR binary operator.

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Perl

A322463 Reverse runs of zeros in binary expansion of n and convert back to decimal.

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Mathematica

PARI

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A285916 Self-inverse permutation: in base 3, reverse the lengths of runs of consecutive equal digits.

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A290078 Where the ratio A235027(n)/n obtains record values.

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PARI

A335858 Nonnegative integers ordered by binary length and then lexicographically by run lengths (considering least significant runs first).

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PARI

Formula

A337672 Numbers with binary expansion Sum_{k = 0..w} b_k * 2^k such that the polynomial Sum_{k = 0..w} (X+k)^2 * (-1)^b_k is constant.

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PARI