A335834 -id:A335834 - cp's OEIS frontend

A335835 Sort the run lengths in binary expansion of n in descending order (with multiplicities).

0, 1, 2, 3, 6, 5, 6, 7, 14, 13, 10, 13, 12, 13, 14, 15, 30, 29, 26, 25, 26, 21, 26, 29, 28, 25, 26, 25, 28, 29, 30, 31, 62, 61, 58, 57, 50, 53, 50, 57, 58, 53, 42, 53, 50, 53, 58, 61, 60, 57, 50, 51, 50, 53, 50, 57, 56, 57, 58, 57, 60, 61, 62, 63, 126, 125

Offset: 0

Rémy Sigrist, Jun 26 2020

This sequence preserves the number of runs (A005811) and the length (A070939) of the binary representation of a number.

			For n = 72:
- the binary representation of 72 is "1001000",
- the corresponding run lengths are: 1, 2, 1, 3,
- in descending order: 3, 2, 1, 1,
- so the binary representation of a(72) is "1110010",
- and a(72) = 114.

Rémy Sigrist, Table of n, a(n) for n = 0..8191
Index entries for sequences related to binary expansion of n

Cf. A005811, A037016 (fixed points), A070939, A101211, A335834.

torl(n) = { my (rr=[]); while (n, my (r=valuation(n+(n%2), 2)); rr = concat(r, rr); n\=2^r); rr }
fromrl(rr) = { my (v=0); for (k=1, #rr, v = (v+(k%2))*2^rr[k]-(k%2)); v }
a(n) = { fromrl(vecsort(torl(n),,4)) }

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

A337242 a(n) is the greatest number m not yet in the sequence such that the binary expansions of m and of n have the same run lengths (up to order but with multiplicity).

Original entry on oeis.org

0, 1, 2, 3, 6, 5, 4, 7, 14, 13, 10, 11, 12, 9, 8, 15, 30, 29, 26, 27, 22, 21, 20, 23, 28, 25, 18, 19, 24, 17, 16, 31, 62, 61, 58, 59, 54, 53, 52, 57, 46, 45, 42, 43, 50, 41, 40, 47, 60, 55, 44, 51, 38, 37, 36, 49, 56, 39, 34, 35, 48, 33, 32, 63, 126, 125, 122

Offset: 0

Rémy Sigrist, Aug 21 2020

This sequence has similarities with A331274; here we consider run lengths in binary expansions, there binary digits.
This sequence is a self-inverse permutation of the nonnegative numbers.
This sequence preserves the number of binary digits (A070939) and the number of runs in binary expansions (A005811).
This sequence has interesting graphical features (see Links section).

			For n = 7280:
- 7280 has binary expansion "1110001110000",
- the corresponding run lengths are: {3, 3, 3, 4},
- there are four numbers k with the same multiset of run lengths:
    k     bin(k)           run lengths
    ----  ---------------  -----------
    7224  "1110000111000"  {3, 4, 3, 3}
    7280  "1110001110000"  {3, 3, 3, 4}
    7288  "1110001111000"  {3, 3, 4, 3}
    7736  "1111000111000"  {4, 3, 3, 3}
- so a(7224) = 7736,
     a(7280) = 7288,
     a(7288) = 7280,
     a(7736) = 7224.

Rémy Sigrist, Table of n, a(n) for n = 0..8191
Rémy Sigrist, Scatterplot of the first 2^20 terms
Rémy Sigrist, Scatterplot of (n-2^19, a(n)-2^19) for n = 2^19..2^20-1
Rémy Sigrist, PARI program for A337242
Index entries for sequences related to binary expansion of n
Index entries for sequences that are permutations of the natural numbers

Cf. A005811, A070939, A101211, A331274, A335834, A335835.

Nest[Function[{a, m}, Append[a, SelectFirst[m, FreeQ[a, #] &]]] @@ {#1, Sort[Map[FromDigits[Join @@ MapIndexed[ConstantArray[Boole[OddQ@ First[#2]], #1] &, #], 2] &, Permutations[Length /@ Split@ IntegerDigits[#2, 2]]], Greater]} & @@ {#, Length@ #} &, {0}, 66] (* Michael De Vlieger, Aug 22 2020 *)

PARI
```
See Links section.
```

a(2^k-1) = 2^k-1 for any k >= 0.

cp's OEIS Frontend

A335835 Sort the run lengths in binary expansion of n in descending order (with multiplicities).

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