A074293 -id:A074293 - cp's OEIS frontend

A074292 Dominant digit in successive groups of 3 from the Kolakoski sequence (A000002).

2, 1, 2, 2, 1, 1, 2, 1, 2, 1, 1, 2, 2, 1, 2, 1, 1, 2, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 2, 1, 2, 2, 1, 1, 2, 1, 1, 2, 2, 1, 2, 1, 1, 2, 1

Offset: 1

Jon Perry, Sep 21 2002

nonn

This appears to be the same as a sequence studied by Claude Lenormand in a letter dated Nov 17 2003: break up the Kolakoski sequence (A000002) into runs of identical symbols and omit one symbol from each run.
The sequence studied by Claude Lenormand is A156257 and is not equal to this one: see A248805 = A156257 - A074292. Differences between the two sequences are at n = 47, 48, 56, 57, 128, 129, 137, 139, 147, 148, 176, 177,... - Jean-Christophe Hervé, Oct 11 2014
As in the Kolakoski sequence, runs in this sequence are of length 1 or 2, because a run XX implies the repetition of exactly the same 3-group in the Kolakoski sequence: -YXX-YXX- or -XXY-XXY- or -XYX-XYX-, and this is not possible 3 times. However, words of the form YXYXY appear in this sequence, but don't appear in the Kolakoski sequence. - Jean-Christophe Hervé, Oct 12 2014

			Kolakoski begins (1,2,2), (1,1,2), (1,2,2), (1,2,2), so this begins 2,1,2,2.

Nathaniel Johnston, Table of n, a(n) for n = 1..10001
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 Apparently unpublished.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_, Oct 02 2018

Cf. A000002, A074293, A074295, A156257, A248805, A318921.

A074292 := proc(n)
    A000002(3*n-2)+A000002(3*n-1)+A000002(3*n)-3 ;
end proc:
seq(A074292(n),n=1..50) ; # R. J. Mathar, Nov 15 2014

OK = {1, 2, 2}; Do[OK = Join[OK, {1+Mod[n-1, 2]}], {n, 3, 1000}, {OK[[n]]}]; If[Count[#, 1] > 1, 1, 2]& /@ Partition[OK, 3]  (* Jean-François Alcover, Nov 13 2014 *)

a(n)=A000002(3n-2)+A000002(3n-1)+A000002(3n)-3. - Benoit Cloitre, Nov 15 2003

More terms from Ray Chandler, Nov 16 2003

Offset corrected by Jean-Christophe Hervé, Oct 11 2014

A074202 Numbers k such that the number of 1's in the binary expansion of k divides 2^k-1.

Original entry on oeis.org

1, 2, 4, 8, 14, 16, 22, 26, 28, 32, 38, 42, 44, 50, 52, 56, 64, 70, 74, 76, 82, 84, 88, 98, 100, 104, 112, 124, 128, 134, 138, 140, 146, 148, 152, 162, 164, 168, 176, 188, 194, 196, 200, 208, 220, 224, 236, 244, 248, 256, 262, 266, 268, 274, 276, 280, 290, 292

Offset: 1

Benoit Cloitre, Sep 17 2002

Odd terms (1, 351, 375, ...) are in A074203.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000120, A000225, A074203, A074293.

Different from A128309.

Select[Range[300], (d = DigitCount[#, 2, 1]) == 1 || PowerMod[2, #, d] == 1 &] (* Amiram Eldar, Jul 30 2020 *)

isok(n) = !((2^n-1) % hammingweight(n)); \\ Michel Marcus, Nov 29 2013

from itertools import count, islice
def A074202_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:not ((1<A074202_list = list(islice(A074202_gen(),20)) # Chai Wah Wu, Mar 09 2023

Edited by N. J. A. Sloane, May 10 2007

A074295 Dominant (i.e., most populous) digit in Kolakoski sequence (A000002) when partitioned into groups of 7.

Original entry on oeis.org

1, 2, 2, 1, 1, 2, 1, 2, 2, 1, 2, 2, 1, 1, 2, 2, 1, 1, 1, 2, 2, 2, 1, 2, 2, 1, 2, 1, 2, 2, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 1, 2, 1, 1, 2, 2, 2, 1, 2, 2, 1, 1, 2, 1, 1, 1, 2, 1

Offset: 1

Jon Perry, Sep 21 2002

			Kolakoski : (1,2,2,1,1,2,1),(2,2,1,2,2,1,1),(2,1,1,2,2,1,2),... hence 1,2,2,...

Nathaniel Johnston, Table of n, a(n) for n = 1..10000

Cf. A074292, A074293.

lim:=400: s:=[1, 2, 2]: for n from 3 to lim do for i from 1 to s[n] do s:=[op(s), 1+((n-1)mod 2)]: od: od: lim2:=floor(nops(s)/7)-1: for n from 0 to lim2 do if(add(s[7*n+k],k=1..7)<=10)then printf("1, "): else printf("2, "): fi: od: # Nathaniel Johnston, May 01 2011

cp's OEIS Frontend

A074292 Dominant digit in successive groups of 3 from the Kolakoski sequence (A000002).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A074202 Numbers k such that the number of 1's in the binary expansion of k divides 2^k-1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Python

Extensions

A074295 Dominant (i.e., most populous) digit in Kolakoski sequence (A000002) when partitioned into groups of 7.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple