A335834 Sort the run lengths in binary expansion of n in ascending order (with multiplicities).

0, 1, 2, 3, 4, 5, 4, 7, 8, 11, 10, 11, 12, 11, 8, 15, 16, 23, 20, 19, 20, 21, 20, 23, 24, 19, 20, 19, 24, 23, 16, 31, 32, 47, 40, 39, 44, 43, 44, 39, 40, 43, 42, 43, 44, 43, 40, 47, 48, 39, 44, 51, 44, 43, 44, 39, 56, 39, 40, 39, 48, 47, 32, 63, 64, 95, 80, 79

Offset: 0

Author

Rémy Sigrist, Jun 26 2020

Keywords

• nonn
• base
• look
• easy

Comments

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

Examples

			For n = 72:
- the binary representation of 72 is "1001000",
- the corresponding run lengths are: 1, 2, 1, 3,
- in ascending order: 1, 1, 2, 3,
- so the binary representation of a(72) is "1011000",
- and a(72) = 88.
		

Links

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

Crossrefs

Cf. A005811, A037014 (fixed points), A070939, A101211, A335835.

Programs

Mathematica

Array[FromDigits[Flatten@ MapIndexed[ConstantArray[Mod[First[#2], 2], #1] &, Sort[Length /@ Split[IntegerDigits[#, 2]]]], 2] &, 67] (* Michael De Vlieger, Jun 27 2020 *)

PARI

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))) }

Formula

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