A318928 Runs-resistance of binary representation of n.

1, 2, 1, 3, 2, 3, 1, 3, 3, 2, 4, 2, 4, 3, 1, 3, 3, 5, 4, 4, 2, 5, 4, 3, 4, 4, 3, 3, 4, 3, 1, 3, 3, 5, 3, 3, 5, 4, 3, 4, 5, 2, 4, 3, 4, 5, 4, 3, 3, 3, 2, 4, 4, 3, 3, 2, 3, 4, 3, 3, 4, 3, 1, 3, 3, 5, 3, 3, 5, 3, 4, 3, 3, 5, 6, 4, 5, 3, 3, 4, 5, 4, 4, 4, 2, 5, 4, 5, 5, 4, 5, 5, 4, 5, 4

Offset

1,2

Comments

Following Lenormand (2003), we define the "runs-resistance" of a finite list L to be the number of times the RUNS transformation must be applied to L in order to reduce L to a list with a single element.

Here it is immaterial whether we read the binary representation of n from left to right or right to left.

The RUNS transformation must be applied at least once, in order to obtain a list, so a(n) >= 1.

Links

N. J. A. Sloane, Table of n, a(n) for n = 1..10000

Claude Lenormand, Deux transformations sur les mots, Preprint, 5 pages, Nov 17 2003. Apparently unpublished. This is a scanned copy of the version that the author sent to me in 2003.

N. J. A. Sloane, Transforms

Example

11 in binary is [1, 0, 1, 1],
which has runs of lengths [1, 1, 2],
which has runs of lengths [2, 1],
which has runs of lengths [1, 1],
which has a single run of length [2].
This took four steps, so a(11) = 4.

Maple

with(transforms);
# compute Lenormand's "resistance" of a list
resist:=proc(a) local ct, i, b;
if whattype(a) <> list then ERROR("input must be a list"); fi:
ct:=0; b:=a; for i from 1 to 100 do
if nops(b)=1 then return(ct); fi;
b:=RUNS(b); ct:=ct+1; od; end;
a:=[1];
for n from 2 to 100 do
b:=convert(n, base, 2);
r:=resist(b);
a:=[op(a), r];
od:

Mathematica

Table[If[n == 1, 1, Length[NestWhileList[Length/@Split[#] &, IntegerDigits[n, 2], Length[#] > 1 &]] - 1], {n, 50}] (* Gus Wiseman, Nov 25 2019 *)

Crossrefs

See A319103 for an inverse, and A319417 and A319418 for records.

Ignoring the first digit gives A329870.

Cuts-resistance is A319416.

Compositions counted by runs-resistance are A329744.

Binary words counted by runs-resistance are A319411 and A329767.

Cf. A107907, A319420, A329860, A329861, A329865, A329867.

Sequence in context: A107337 A066376 A151682 * A159918 A278573 A108663

Adjacent sequences: A318925 A318926 A318927 * A318929 A318930 A318931

Keyword

nonn,base,nice

Author

N. J. A. Sloane, Sep 09 2018

Extensions

a(1) corrected by N. J. A. Sloane, Sep 20 2018

Status

approved