A319418 -id:A319418 - cp's OEIS frontend

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

N. J. A. Sloane, Sep 09 2018

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.

			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.

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 Apparently unpublished.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

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.

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:

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

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

A319103 a(n) is the least k > 0 such that A318928(k) = n.

Original entry on oeis.org

1, 3, 2, 4, 11, 18, 75, 621, 9638, 1264052, 1294752365, 20699153586797, 43409394810283725529

Offset: 0

Rémy Sigrist, Sep 10 2018

This sequence is well defined and infinite:
- for any n > 1, we can build a number m such that A318928(m) = 1 + A318928(n),
- let (b_1, ..., b_k) be the binary representation of n,
- let r_1 = 1, and for i = 1..k-1: r_{i+1} = r_i if b_{i+1} = b_i and r_{i+1} = 2 - r_i otherwise,
- the number m whose run lengths in binary representation are (r_1, ..., r_k) satisfies A318928(m) = 1 + A318928(n).
a(11) <= 42414573279593.
Here A318928(1) is considered to be 0, which differs from the current definition of A318928. However, I think it is quite natural to define A318928(1) to be 0. - Hiroaki Yamanouchi, Sep 22 2018

			The first terms of A318928, alongside the corresponding terms in this sequence, are:
  n   A318928(n)  Corresponding terms
  --  ----------  -------------------
   1           0  a(0) = 1
   2           2  a(2) = 2
   3           1  a(1) = 3
   4           3  a(3) = 4
   5           2
   6           3
   7           1
   8           3
   9           3
  10           2
  11           4  a(4) = 11
  12           2
  13           4

Rémy Sigrist, PARI program for A319103

Cf. A318928.

See A319417, A319418 for record values in A318928.

PARI
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See Links section.
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a(11)-a(12) from Hiroaki Yamanouchi, Sep 22 2018

A319417 Records in A318928.

Original entry on oeis.org

1, 2, 4, 11, 18, 75, 621, 9638, 1264052, 1294752365, 20699153586797, 43409394810283725529

Offset: 1

N. J. A. Sloane, Sep 21 2018

It is very plausible that the next two terms are 1264052 and 1294752365, as in Rémy Sigrist's A319103, but that needs to be checked.

Cf. A318928, A319103, A319418 (a(n) written in base 2).

a(9)-a(10) from Rémy Sigrist's results, Sep 25 2018

a(11)-a(12) from Hiroaki Yamanouchi, Sep 25 2018

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A318928 Runs-resistance of binary representation of n.

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A319103 a(n) is the least k > 0 such that A318928(k) = n.

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A319417 Records in A318928.

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