A054354 -id:A054354 - cp's OEIS frontend

A376604 Second differences of the Kolakoski sequence (A000002). First differences of A054354.

-1, -1, 1, 1, -2, 2, -1, -1, 2, -1, -1, 1, 1, -2, 1, 1, -1, -1, 2, -2, 1, 1, -2, 2, -1, -1, 1, 1, -2, 1, 1, -2, 2, -1, -1, 2, -1, -1, 1, 1, -2, 2, -1, -1, 2, -2, 1, 1, -2, 1, 1, -1, -1, 2, -1, -1, 1, 1, -2, 2, -1, -1, 2, -1, -1, 1, 1, -2, 1, 1, -2, 2, -1, -1

Offset: 1

Gus Wiseman, Oct 02 2024

sign

Since A000002 has no runs of length 3, this sequence contains no zeros.

The densities appear to approach (1/3, 1/3, 1/6, 1/6).

			The Kolakoski sequence (A000002) is:
  1, 2, 2, 1, 1, 2, 1, 2, 2, 1, 2, 2, 1, 1, 2, 1, 1, 2, 2, 1, 2, 1, 1, 2, 1, 2, 2, ...
with first differences (A054354):
  1, 0, -1, 0, 1, -1, 1, 0, -1, 1, 0, -1, 0, 1, -1, 0, 1, 0, -1, 1, -1, 0, 1, -1, ...
with first differences (A376604):
  -1, -1, 1, 1, -2, 2, -1, -1, 2, -1, -1, 1, 1, -2, 1, 1, -1, -1, 2, -2, 1, 1, -2, ...

A001462 is Golomb's sequence.
A078649 appears to be zeros of the first and third differences.
A288605 gives positions of first appearances of each balance.
A306323 gives a 'broken' version.
A333254 lists run-lengths of differences between consecutive primes.
For the Kolakoski sequence (A000002):
- Restrictions: A074264, A100428, A100429, A156263, A156264.
- Patterns: A013947, A013948, A022292, A074262, A074263, A156242, A156243.
- Statistics: A054353, A074286, A088568, A156077, A156253.
- Transformations: A054354, A156728, A332273, A332875, A333229, A376604.
For second differences: A036263 (prime), A073445 (composite), A376559 (perfect-power), A376562 (non-perfect-power), A376590 (squarefree), A376593 (nonsquarefree), A376596 (prime-power), A376599 (non-prime-power).

kolagrow[q_]:=If[Length[q]<2,Take[{1,2},Length[q]+1],Append[q,Switch[{q[[Length[Split[q]]]],q[[-2]],Last[q]},{1,1,2},1,{1,2,1},2,{2,1,1},2,{2,1,2},2,{2,2,1},1,{2,2,2},1]]]
kol[n_]:=Nest[kolagrow,{1},n-1];
Differences[kol[100],2]

A156728 a(n) = abs(A054354(n)).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1

Offset: 1

Benoit Cloitre, Feb 14 2009

nonn

This sequence is the image of the Kolakoski sequence A000002 under the morphism 1->1, 2->01. - Gabriele Fici, Aug 12 2013

Jean-Marc Fedou and Gabriele Fici, Some remarks on differentiable sequences and recursivity, Journal of Integer Sequences 13(3): Article 10.3.2 (2010).

Cf. A000002, A054354, A156251, A156253.

a2 = {1, 2, 2}; Do[ a2 = Join[a2, {1 + Mod[n - 1, 2]}], {n, 3, 105}, {i, 1, a2[[n]]}]; a[n_] := Mod[a2[[n]] + a2[[n + 1]], 2]; Table[a[n], {n, 1, 105}] (* Jean-François Alcover, Jun 18 2013 *)

a(n) = (v(n+1) - v(n) + 1)/2 where v(n) = A156253(n) - A156251(n).
a(n) = (A000002(n) + A000002(n+1)) mod 2.
a(n) = A156253(n+1) - A156253(n). - Alan Michael Gómez Calderón, Dec 20 2024

A000002 Kolakoski sequence: a(n) is length of n-th run; a(1) = 1; sequence consists just of 1's and 2's.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

Historical note: the sequence might be better called the Oldenburger-Kolakoski sequence, since it was discussed by Rufus Oldenburger in 1939; see links. - Clark Kimberling, Dec 06 2012.  However, to avoid confusion, this sequence will be known in the OEIS as the Kolakoski sequence. It is undesirable to have some entries refer to the Oldenburger-Kolakoski sequence and others to the Kolakoski sequence. - N. J. A. Sloane, Nov 22 2017
It is an unsolved problem to show that the density of 1's is equal to 1/2.
A weaker problem is to construct a combinatorial bijection between the set of positions of 1's and the set of positions of 2's. - Gus Wiseman, Mar 01 2016
The sequence is cubefree and all square subwords have lengths which are one of 2, 4, 6, 18 and 54 (see A294447) [Carpi, 1994].
This is a fractal sequence: replace each run with its length and recover the original sequence. - Kerry Mitchell, Dec 08 2005
Kupin and Rowland write: We use a method of Goulden and Jackson to bound freq_1(K), the limiting frequency of 1 in the Kolakoski word K. We prove that |freq_1(K) - 1/2| <= 17/762, assuming the limit exists and establish the semirigorous bound |freq_1(K) - 1/2| <= 1/46. - Jonathan Vos Post, Sep 16 2008
freq_1(K) is conjectured to be 1/2 + O(log(K)) (see PlanetMath link). - Jon Perry, Oct 29 2014
Conjecture: Taking the sequence in word lengths of 10, for example, batch 1-10, 11-20, etc., then there can only be 4, 5 or 6 1's in each batch. - Jon Perry, Sep 26 2012
From Jean-Christophe Hervé, Oct 04 2014: (Start)
The sequence does not contain words of the form ababa, because this would imply the impossible 111 (1 b, 1 a, 1 b) somewhere before. This demonstrates the conjecture made by Jon Perry: more than 6 1's or 6 2's in a word of 10 would necessitate something like aabaabaaba, which would imply the impossible 12121 before (word aabaababaa is also impossible because of ababa). The remark on the sextuplets below even shows that the number of 1's in any 9-tuplet is always 4 or 5.
There are only 6 triples that appear in the sequence (112, 121, 122, 211, 212 and 221); and by the preceding argument, only 18 sextuplets: the 6 double triples (112112, etc.); 112122, 112212, 121122, 121221, 211212, and 211221; and those obtained by reversing the order of the triples (122112, etc.). Regarding the density of 1's in the sequence, these 12 sextuplets all have a density 1/2 of 1's, and the 6 double triples all lead to a word with this exact density after transformation by the Kolakoski rules, for example: 112112 -> 12112122 (4 1's/8); this is because the second triple reverses the numbers of 1's and 2's generated by the first triple. Therefore, the sequence can be split into the double triples on one side, a part whose transformation (which is in the sequence) has a density of 1's of 1/2; and a part with the other sextuplets, which has directly the same density of 1's. (End)
If we map 1 to +1 and 2 to -1, then the mapped sequence would have a [conjectured] mean of 0, since the Kolakoski sequence is [conjectured] to have an equal density (1/2) of 1s and 2s. For the partial sums of this mapped sequence, see A088568. - Daniel Forgues, Jul 08 2015
Looking at the plot for A088568, it seems that although the asymptotic densities of 1s and 2s appear to be 1/2, there might be a bias in favor of the 2s. I.e., D(1) = 1/2 - O(log(n)/n), D(2) = 1/2 + O(log(n)/n). - Daniel Forgues, Jul 11 2015
From Michel Dekking, Jan 31 2018: (Start)
(a(n)) is the unique fixed point of the 2-block substitution beta
    11 -> 12
    12 -> 122
    21 -> 112
    22 -> 1122.
A 2-block substitution beta maps a word w(1)...w(2n) to the word
    beta(w(1)w(2))...beta(w(2n-1)w(2n)).
If the word has odd length, then the last letter is ignored.
It was noted by me in 1979 in the Bordeaux seminar on number theory that (a(n+1)) is fixed point of the 2-block substitution 11 -> 21, 12 -> 211, 21 -> 221,  22 -> 2211. (End)
Named after the American artist and recreational mathematician William George Kolakoski (1944-1997). - Amiram Eldar, Jun 17 2021

			Start with a(1) = 1. By definition of the sequence, this says that the first run has length 1, so it must be a single 1, and a(2) = 2. Thus, the second run (which starts with this 2) must have length 2, so the third term must be also be a(3) = 2, and the fourth term can't be a 2, so must be a(4) = 1. Since a(3) = 2, the third run must have length 2, so we deduce a(5) = 1, a(6) = 2, and so on. The correction I made was to change a(4) to a(5) and a(5) to a(6). - _Labos Elemer_, corrected by _Graeme McRae_

Jean-Paul Allouche and Jeffrey Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 337.
Éric Angelini, "Jeux de suites", in Dossier Pour La Science, pp. 32-35, Volume 59 (Jeux math'), April/June 2008, Paris.
F. M. Dekking, What Is the Long Range Order in the Kolakoski Sequence?, in The mathematics of long-range aperiodic order (Waterloo, ON, 1995), 115-125, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 489, Kluwer Acad. Publ., Dordrecht, 1997. Math. Rev. 98g:11022.
Michael S. Keane, Ergodic theory and subshifts of finite type, Chap. 2 of T. Bedford et al., eds., Ergodic Theory, Symbolic Dynamics and Hyperbolic Spaces, Oxford, 1991, esp. p. 50.
J. C. Lagarias, Number Theory and Dynamical Systems, pp. 35-72 of S. A. Burr, ed., The Unreasonable Effectiveness of Number Theory, Proc. Sympos. Appl. Math., 46 (1992). Amer. Math. Soc.
Michel Rigo, Formal Languages, Automata and Numeration Systems, 2 vols., Wiley, 2014. Mentions this sequence - see "List of Sequences" in Vol. 2.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Ilan Vardi, Computational Recreations in Mathematica. Addison-Wesley, Redwood City, CA, 1991, p. 233.

N. J. A. Sloane, Table of n, a(n) for n = 1..10502
Jean-Paul Allouche, Michael Baake, Julien Cassaigne and David Damanik, Palindrome complexity, arXiv:math/0106121 [math.CO], 2001; Theoretical Computer Science, Vol. 292 (2003), pp. 9-31.
Michael Baake and Bernd Sing, Kolakoski-(3,1) is a (deformed) model set, arXiv:math/0206098 [math.MG], 2002-2003.
Alex Bellos and Brady Haran, The Kolakoski Sequence, Numberphile video (2017).
Olivier Bordellès and Benoit Cloitre, Bounds for the Kolakoski Sequence, J. Integer Sequences, Vol. 14 (2011), Article 11.2.1.
Richard P. Brent, Fast algorithms for the Kolakoski sequence, Slides from a talk, 2016.
Éric Brier, Rémi Géraud-Stewart, David Naccache, Alessandro Pacco, and Emanuele Troiani, The Look-and-Say The Biggest Sequence Eventually Cycles, arXiv:2006.07246 [math.DS], 2020.
Arturo Carpi, On repeated factors in C^infinity-words, Information Processing Letters, Vol. 52 (1994), pp. 289-294.
Benoit Cloitre, The Kolakoski transform of words.
Benoit Cloitre, Plot of walk on the first 60000 terms (step of unit length moving right with angle Pi/2 if 2 and left with angle -Pi/2 if 1 starting at (0,0)).
F. M. Dekking, Regularity and irregularity of sequences generated by automata, Seminar on Number Theory, 1979-1980 (Talence, 1979-1980), Exp. No. 9, 10 pp., Univ. Bordeaux I, Talence, 1980.
F. M. Dekking, On the structure of self-generating sequences, Seminar on Number Theory, 1980-1981 (Talence, 1980-1981), Exp. No. 31, 6 pp., Univ. Bordeaux I, Talence, 1981. Math. Rev. 83e:10075. JSTOR
F. M. Dekking, What Is the Long Range Order in the Kolakoski Sequence?, Report 95-100, Technische Universiteit Delft, 1995.
F. M. Dekking, The Thue-Morse Sequence in Base 3/2, J. Int. Seq., Vol. 26 (2023), Article 23.2.3.
Jörg Endrullis, Dimitri Hendriks and Jan Willem Klop, Degrees of Streams, Integers, Vol. 11B (2011), A6.
David Eppstein, The Kolakoski tree and The Kolakoski sequence via bit tricks instead of recursion.
Jean-Marc Fédou and Gabriele Fici, Some remarks on differentiable sequences and recursivity, Journal of Integer Sequences, Vol. 13 (2010), Article 10.3.2.
Abdallah Hammam, Some new Formulas for the Kolakoski Sequence A000002, Turkish Journal of Analysis and Number Theory, Vol. 4, No. 3 (2016), pp. 54-59.
Mari Huova and Juhani Karhumäki, On Unavoidability of k-abelian Squares in Pure Morphic Words, Journal of Integer Sequences, Vol. 16 (2013), #13.2.9.
Clark Kimberling, Integer Sequences and Arrays, Illustration of the Kolakoski sequence.
William Kolakoski, Problem 5304, Amer. Math. Monthly, Vol. 72, No. 8 (1965), p. 674; Self Generating Runs, Solution to Problem 5304 by Necdet Üçoluk, Vol. 73, No. 6 (1966), pp. 681-682.
Leonid V. Kovalev, Kolakoski sequence II.
Elizabeth J. Kupin and Eric S. Rowland, Bounds on the frequency of 1 in the Kolakoski word, arXiv:0809.2776 [math.CO], Sep 16, 2008.
Rabie A. Mahmoud, Hardware Implementation of Binary Kolakoski Sequence, Research Gate, 2015.
Kerry Mitchell, Spirolateral-Type Images from Integer Sequences, 2013.
Johan Nilsson, A Space Efficient Algorithm for the Calculation of the Digit Distribution in the Kolakoski Sequence, J. Int. Seq., Vol. 15 (2012), Article 12.6.7; arXiv preprint, arXiv:1110.4228 [math.CO], Oct 19, 2011.
J. Nilsson, Letter Frequencies in the Kolakoski Sequence, Acta Physica Polonica A, Vol. 126 (2014), pp. 549-552.
Rufus Oldenburger, Exponent trajectories in symbolic dynamics, Trans. Amer. Math. Soc., Vol. 46 (1939), pp. 453-466.
Matthew Parker, The first 50K terms (7-Zip compressed file).
Gheorghe Păun and Arto Salomaa, Self-reading sequences, Amer. Math. Monthly, Vol. 103, No. 2 (1996), pp. 166-168.
Michael Rao, Trucs et bidules sur la séquence de Kolakoski, 2012, in French.
A. Scolnicov, Kolakoski sequence, PlanetMath.org.
Bobby Shen, A uniformness conjecture of the Kolakoski sequence, graph connectivity, and correlations, arXiv:1703.00180 [math.CO], 2017.
Bernd Sing, More Kolakoski Sequences, INTEGERS, Vol. 11B (2011), A14.
N. J. A. Sloane, Handwritten notes on Self-Generating Sequences, 1970. (note that A1148 has now become A005282)
N. J. A. Sloane, Coordination Sequences, Planing Numbers, and Other Recent Sequences (II), Experimental Mathematics Seminar, Rutgers University, Jan 31 2019, Part I, Part 2, Slides. (Mentions this sequence)
Bertran Steinsky, A Recursive Formula for the Kolakoski Sequence A000002, J. Integer Sequences, Vol. 9 (2006), Article 06.3.7.
Eric Weisstein's World of Mathematics, Kolakoski Sequence
Wikipedia, Kolakoski sequence.
Gus Wiseman, Kolakoski fractal animation for n=40000.
Ed Wynn, Program to generate A000002 in Shakespeare Programming Language.
Index entries for "core" sequences

Cf. A001083, A006928, A042942, A069864, A010060, A078929, A171899, A054353 (partial sums), A074286, A216345, A294447.
Cf. A054354, bisections: A100428, A100429.
Cf. A013947, A156077, A234322 (positions, running total and percentage of 1's).
Cf. A118270.
Cf. A049705, A088569 (are either subsequences of A000002? - Jon Perry, Oct 30 2014)
Kolakoski-type sequences using other seeds than (1,2):
A078880 (2,1), A064353 (1,3), A071820 (2,3), A074804 (3,2), A071907 (1,4), A071928 (2,4), A071942 (3,4), A074803 (4,2), A079729 (1,2,3), A079730 (1,2,3,4).
Other self-describing: A001462 (Golomb sequence, see also references therein), A005041, A100144.
Cf. A088568 (partial sums of [3 - 2 * a(n)]).

a = 1:2: drop 2 (concat . zipWith replicate a . cycle $ [1,2]) -- John Tromp, Apr 09 2011

M := 100; s := [ 1,2,2 ]; for n from 3 to M do for i from 1 to s[ n ] do s := [ op(s),1+((n-1)mod 2) ]; od: od: s; A000002 := n->s[n];
# alternative implementation based on the Cloitre formula:
A000002 := proc(n)
    local ksu,k ;
    option remember;
    if n = 1 then
        1;
    elif n <=3 then
        2;
    else
        for k from 1 do
            ksu := add(procname(i),i=1..k) ;
            if n = ksu then
                return (3+(-1)^k)/2 ;
            elif n = ksu+ 1 then
                return (3-(-1)^k)/2 ;
            end if;
        end do:
    end if;
end proc: # R. J. Mathar, Nov 15 2014

a[steps_] := Module[{a = {1, 2, 2}}, Do[a = Append[a, 1 + Mod[(n - 1), 2]], {n, 3, steps}, {i, a[[n]]}]; a]
a[ n_] := If[ n < 3, Max[ 0, n], Module[ {an = {1, 2, 2}, m = 3}, While[ Length[ an] < n, an = Join[ an, Table[ Mod[m, 2, 1], { an[[ m]]} ]]; m++]; an[[n]]]] (* Michael Somos, Jul 11 2011 *)
n=8; Prepend[ Nest[ Flatten[ Partition[#, 2] /. {{2, 2} -> {2, 2, 1, 1}, {2, 1} -> {2, 2, 1}, {1, 2} -> {2, 1, 1}, {1, 1} -> {2, 1}}] &, {2, 2}, n], 1] (* Birkas Gyorgy, Jul 10 2012 *)
KolakoskiSeq[n_Integer] := Block[{a = {1, 2, 2}}, Fold[Join[#1, ConstantArray[Mod[#2, 2, 1], #1[[#2]]]] &, a, Range[3, n]]]; KolakoskiSeq[999] (* Mikk Heidemaa, Nov 01 2024 *) (* Corrected by Giorgos Kalogeropoulos, May 09 2025 *)

my(a=[1,2,2]); for(n=3,80, for(i=1,a[n],a=concat(a,2-n%2))); a

{a(n) = local(an=[1, 2, 2], m=3); if( n<1, 0, while( #an < n, an = concat( an, vector(an[m], i, 2-m%2)); m++); an[n])};

# For explanation see link.
def Kolakoski():
    x = y = -1
    while True:
        yield [2,1][x&1]
        f = y &~ (y+1)
        x ^= f
        y = (y+1) | (f & (x>>1))
K = Kolakoski()
print([next(K) for  in range(100)]) # _David Eppstein, Oct 15 2016

These two formulas define completely the sequence: a(1)=1, a(2)=2, a(a(1) + a(2) + ... + a(k)) = (3 + (-1)^k)/2 and a(a(1) + a(2) + ... + a(k) + 1) = (3 - (-1)^k)/2. - Benoit Cloitre, Oct 06 2003
a(n+2)*a(n+1)*a(n)/2 = a(n+2) + a(n+1) + a(n) - 3 (this formula doesn't define the sequence, it is just a consequence of the definition). - Benoit Cloitre, Nov 17 2003
a(n+1) = 3 - a(n) + (a(n) - a(n-1))*(a(b(n)) - 1), where b(n) is the sequence A156253. - Jean-Marc Fedou and Gabriele Fici, Mar 18 2010
a(n) = (3 + (-1)^A156253(n))/2. - Benoit Cloitre, Sep 17 2013
Conjectures from Boštjan Gec, Oct 07 2024: (Start)
a(n)*(a(n-1) + a(n-2) - 3) + a(n-1)*a(n-2) + 7 = 3*a(n-1) + 3*a(n-2).
a(n)*(a(n-1) + a(n-2) - 3) = a(n-3)*(a(n-1) + a(n-2) - 3). (End)
Comment from Kevin Ryde, Oct 07 2024: The above formulas are true: The parts identify when terms are same or different and they hold for any sequence of 1's and 2's with run lengths 1 or 2.

Minor edits to example and PARI code made by M. F. Hasler, May 07 2014

A376593 Second differences of consecutive nonsquarefree numbers (A013929). First differences of A078147.

Original entry on oeis.org

-3, 2, 1, -2, 0, 2, -3, 1, -1, 3, 0, 0, 0, -3, 2, -2, 0, 1, 0, 0, 2, -1, -2, 3, 0, -1, -2, 3, -3, 2, 1, -2, 0, 2, -2, -1, 0, 3, 0, 0, 0, -3, 2, -2, 2, -2, 0, 1, 2, -1, -2, 3, 0, -1, -2, 1, 0, -1, 2, 1, -2, 0, 2, -3, 1, -1, 2, -2, 3, 0, 0, -3, 2, 1, -2, 0, 2

Offset: 1

Gus Wiseman, Oct 01 2024

sign

The range is {-3, -2, -1, 0, 1, 2, 3}.

			The nonsquarefree numbers (A013929) are:
  4, 8, 9, 12, 16, 18, 20, 24, 25, 27, 28, 32, 36, 40, 44, 45, 48, 49, 50, 52, ...
with first differences (A078147):
  4, 1, 3, 4, 2, 2, 4, 1, 2, 1, 4, 4, 4, 4, 1, 3, 1, 1, 2, 2, 2, 4, 3, 1, 4, 4, ...
with first differences (A376593):
  -3, 2, 1, -2, 0, 2, -3, 1, -1, 3, 0, 0, 0, -3, 2, -2, 0, 1, 0, 0, 2, -1, -2, ...

The version for A000002 is A376604, first differences of A054354.
The first differences were A078147.
Zeros are A376594, complement A376595.
A000040 lists the prime numbers, differences A001223.
A005117 lists squarefree numbers, differences A076259.
A064113 lists positions of adjacent equal prime gaps.
A114374 counts partitions into nonsquarefree numbers.
A246655 lists prime-powers exclusive, inclusive A000961.
A333254 lists run-lengths of differences between consecutive primes.
For second differences: A036263 (prime), A073445 (composite), A376559 (perfect-power), A376562 (non-perfect-power), A376590 (squarefree), A376596 (prime-power inclusive), A376599 (non-prime-power inclusive).
For nonsquarefree numbers: A013929 (terms), A078147 (first differences), A376594 (inflections and undulations), A376595 (nonzero curvature).
Cf. A061398, A053797, A053806, A120992, A182853, A251092, A373198, A375707, A376305, A376306, A376311, A376312, A376655.

Differences[Select[Range[100],!SquareFreeQ[#]&],2]

from math import isqrt
from sympy import mobius, factorint
def A376593(n):
    def f(x): return n+sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    k = next(i for i in range(1,5) if any(d>1 for d in factorint(m+i).values()))
    return next(i for i in range(1-k,5-k) if any(d>1 for d in factorint(m+(k<<1)+i).values())) # Chai Wah Wu, Oct 02 2024

A376596 Second differences of consecutive prime-powers inclusive (A000961). First differences of A057820.

Original entry on oeis.org

0, 0, 0, 1, -1, 0, 1, 0, 1, -2, 1, 2, -2, 0, 0, 0, -1, 4, -1, -2, 2, -2, 2, 2, -4, 1, 0, 1, -2, 4, -4, 0, 4, 2, -4, -2, 2, -2, 2, 4, -4, -2, -1, 2, 3, -4, 8, -8, 4, 0, -2, -2, 2, 2, -4, 8, -8, 2, -2, 10, 0, -8, -2, 2, 2, -4, 0, 6, -3, -4, 5, 0, -4, 4, -2, -2

Offset: 1

Gus Wiseman, Oct 02 2024

sign

For the exclusive version, shift left once.

			The prime-powers inclusive (A000961) are:
  1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, ...
with first differences (A057820):
  1, 1, 1, 1, 2, 1, 1, 2, 2, 3, 1, 2, 4, 2, 2, 2, 2, 1, 5, 4, 2, 4, 2, 4, 6, 2, 3, ...
with first differences (A376596):
  0, 0, 0, 1, -1, 0, 1, 0, 1, -2, 1, 2, -2, 0, 0, 0, -1, 4, -1, -2, 2, -2, 2, 2, ...

The version for A000002 is A376604, first differences of A054354.
For first differences we had A057820, sorted firsts A376340(n)+1 (except first term).
Positions of zeros are A376597, complement A376598.
Sorted positions of first appearances are A376653, exclusive A376654.
A000961 lists prime-powers inclusive, exclusive A246655.
A001597 lists perfect-powers, complement A007916.
A023893 and A023894 count integer partitions into prime-powers, factorizations A000688.
A064113 lists positions of adjacent equal prime gaps.
For prime-powers inclusive: A057820 (first differences), A376597 (inflections and undulations), A376598 (nonzero curvature).
For second differences: A036263 (prime), A073445 (composite), A376559 (perfect-power), A376562 (non-perfect-power), A376590 (squarefree), A376593 (nonsquarefree), A376599 (non-prime-power).
Cf. A025475, A045542, A053707, A069623, A093555, A174965, A251092, A333254, A336416, A361102.

Differences[Select[Range[1000],#==1||PrimePowerQ[#]&],2]

from sympy import primepi, integer_nthroot
def A376596(n):
    def iterfun(f,n=0):
        m, k = n, f(n)
        while m != k: m, k = k, f(k)
        return m
    def f(x): return int(n+x-1-sum(primepi(integer_nthroot(x,k)[0]) for k in range(1,x.bit_length())))
    return (a:=iterfun(f,n))-((b:=iterfun(lambda x:f(x)+1,a))<<1)+iterfun(lambda x:f(x)+2,b) # Chai Wah Wu, Oct 02 2024

A376562 Second differences of consecutive non-perfect-powers (A007916). First differences of A375706.

Original entry on oeis.org

1, -1, 0, 2, -2, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 1, 0, -1, 0, 0, 1, -1, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Oct 01 2024

sign

Non-perfect-powers (A007916) are numbers without a proper integer root.

			The non-perfect powers (A007916) are:
  2, 3, 5, 6, 7, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 26, 28, ...
with first differences (A375706):
  1, 2, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, ...
with first differences (A376562):
  1, -1, 0, 2, -2, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 1, 0, -1, 0, 0, 1, -1, 0, ...

The version for A000002 is A376604, first differences of A054354.
For first differences we had A375706, ones A375740, complement A375714.
Positions of zeros are A376588, complement A376589.
Runs of non-perfect-powers:
- length: A375702 = A053289(n+1) - 1
- first: A375703 (same as A216765 with 2 exceptions)
- last: A375704 (same as A045542 with 8 removed)
- sum: A375705
A000961 lists prime-powers inclusive, exclusive A246655.
A007916 lists non-perfect-powers, complement A001597.
A112344 counts integer partitions into perfect-powers, factorizations A294068.
A333254 gives run-lengths of differences between consecutive primes.
For non-perfect-powers: A375706 (first differences), A376588 (inflections and undulations), A376589 (nonzero curvature).
For second differences: A036263 (prime), A073445 (composite), A376559 (perfect-power), A376590 (squarefree), A376593 (nonsquarefree), A376596 (prime-power inclusive), A376599 (non-prime-power inclusive).
Cf. A025475, A052410, A053707, A064113, A069623, A093555, A174965, A182853, A336416, A336417, A361102.

radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
Differences[Select[Range[100],radQ],2]

from itertools import count
from sympy import mobius, integer_nthroot, perfect_power
def A376562(n):
    def f(x): return int(n+1-sum(mobius(k)*(integer_nthroot(x, k)[0]-1) for k in range(2, x.bit_length())))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    r = m+((k:=next(i for i in count(1) if not perfect_power(m+i)))<<1)
    return next(i for i in count(1-k) if not perfect_power(r+i)) # Chai Wah Wu, Oct 02 2024

A376590 Second differences of consecutive squarefree numbers (A005117). First differences of A076259.

Original entry on oeis.org

0, 1, -1, 0, 2, -2, 1, -1, 0, 1, 0, 0, -1, 0, 2, 0, -2, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 2, -2, 3, -2, 0, 0, -1, 0, 1, -1, 2, -2, 0, 1, -1, 0, 1, -1, 2, -2, 0, 2, -2, 1, -1, 0, 1, 0, 0, -1, 0, 1, 2, -3, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 2, -2, 2, -2, 3, -2, -1

Offset: 1

Gus Wiseman, Oct 01 2024

sign

			The squarefree numbers (A005117) are:
  1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, ...
with first differences (A076259):
  1, 1, 2, 1, 1, 3, 1, 2, 1, 1, 2, 2, 2, 1, 1, 3, 3, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, ...
with first differences (A376590):
  0, 1, -1, 0, 2, -2, 1, -1, 0, 1, 0, 0, -1, 0, 2, 0, -2, 0, 1, -1, 0, 1, -1, 0, 1, ...

The version for A000002 is A376604, first differences of A054354.
The first differences were A076259, see also A375927, A376305, A376306, A376307, A376311.
Zeros are A376591, complement A376592.
Sorted positions of first appearances are A376655.
A000040 lists the prime numbers, differences A001223.
A001597 lists perfect-powers, complement A007916.
A005117 lists squarefree numbers, complement A013929 (differences A078147).
A073576 counts integer partitions into squarefree numbers, factorizations A050320.
A333254 lists run-lengths of differences between consecutive primes.
For second differences: A036263 (prime), A073445 (composite), A376559 (perfect-power), A376562 (non-perfect-power), A376593 (nonsquarefree), A376596 (prime-power inclusive), A376599 (non-prime-power inclusive).
For squarefree numbers: A076259 (first differences), A376591 (inflections and undulations), A376592 (nonzero curvature), A376655 (sorted first positions).
Cf. A000961, A007674, A053797, A053806, A061398, A072284, A112925, A112926, A120992, A251092, A373198, A376342.

Differences[Select[Range[100],SquareFreeQ],2]

from math import isqrt
from sympy import mobius
def A376590(n):
    def iterfun(f,n=0):
        m, k = n, f(n)
        while m != k: m, k = k, f(k)
        return m
    def f(x): return n+x-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    a = iterfun(f,n)
    b = iterfun(lambda x:f(x)+1,a)
    return a+iterfun(lambda x:f(x)+2,b)-(b<<1) # Chai Wah Wu, Oct 02 2024

A376559 Second differences of consecutive perfect powers (A001597). First differences of A053289.

Original entry on oeis.org

1, -3, 6, 2, -7, 3, -1, 9, 2, 2, 2, 2, -17, -1, 13, 9, 2, -7, -11, 9, -5, 20, 2, -16, -1, 21, 2, 2, -15, -11, 30, 2, 2, 2, 2, 2, 2, 2, -22, -15, 41, 2, 2, 2, -36, 3, 37, 2, 2, 2, -34, -11, 49, 2, 2, -66, 45, 3, -61, 2, 83, 2, 2, 2, 2, -63, 25, 42, 2, -9, -89

Offset: 1

Gus Wiseman, Sep 28 2024

sign

Perfect-powers A007916 are numbers with a proper integer root.

Does this sequence contain zero?

			The perfect powers (A001597) are:
  1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81, 100, 121, 125, 128, 144, 169, 196, ...
with first differences (A053289):
  3, 4, 1, 7, 9, 2, 5, 4, 13, 15, 17, 19, 21, 4, 3, 16, 25, 27, 20, 9, 18, 13, ...
with first differences (A376559):
  1, -3, 6, 2, -7, 3, -1, 9, 2, 2, 2, 2, -17, -1, 13, 9, 2, -7, -11, 9, -5, 20, ...

The version for A000002 is A376604, first differences of A054354.
For first differences we have A053289, union A023055, firsts A376268, A376519.
A000961 lists prime-powers inclusive, exclusive A246655.
A001597 lists perfect-powers, complement A007916.
A112344 counts integer partitions into perfect-powers, factorizations A294068.
For perfect-powers: A053289 (first differences), A376560 (positive curvature), A376561 (negative curvature).
For second differences: A036263 (prime), A073445 (composite), A376562 (non-perfect-power), A376590 (squarefree), A376593 (nonsquarefree), A376596 (prime-power), A376599 (non-prime-power).
Cf. A045542, A052410, A053707, A064113, A069623, A174965, A216765, A251092, A333254, A336416, A361102.

perpowQ[n_]:=n==1||GCD@@FactorInteger[n][[All,2]]>1;
Differences[Select[Range[1000],perpowQ],2]

lista(nn) = my(v = concat (1, select(ispower, [1..nn])), w = vector(#v-1, i, v[i+1] - v[i])); vector(#w-1, i, w[i+1] - w[i]); \\ Michel Marcus, Oct 02 2024

from sympy import mobius, integer_nthroot
def A376559(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return int(n-1+x+sum(mobius(k)*(integer_nthroot(x,k)[0]-1) for k in range(2,x.bit_length())))
    a = bisection(f,n,n)
    b = bisection(lambda x:f(x)+1,a,a)
    return a+bisection(lambda x:f(x)+2,b,b)-(b<<1) # Chai Wah Wu, Oct 02 2024

A376599 Second differences of consecutive non-prime-powers inclusive (A024619). First differences of A375735.

Original entry on oeis.org

-2, 0, -1, 2, -1, -1, 0, 1, 0, 0, 0, 1, -2, 0, 0, 1, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, -1, 0, 0, 0, 1, 0, -1, 1, -1, 1, -1, 0, 1, 0, -1, 0, 0, 0, 1, 0, 0, -1, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 1, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, -1, 0, 0, 0, 0, 0, 1, -1, 0

Offset: 1

Gus Wiseman, Oct 02 2024

sign

Inclusive means 1 is a prime-power but not a non-prime-power. For the exclusive version, shift left once.

			The non-prime-powers inclusive (A024619) are:
  6, 10, 12, 14, 15, 18, 20, 21, 22, 24, 26, 28, 30, 33, 34, 35, 36, 38, 39, 40, ...
with first differences (A375735):
  4, 2, 2, 1, 3, 2, 1, 1, 2, 2, 2, 2, 3, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, ...
with first differences (A376599):
  -2, 0, -1, 2, -1, -1, 0, 1, 0, 0, 0, 1, -2, 0, 0, 1, -1, 0, 1, 0, -1, 0, 1, 0, ...

The version for A000002 is A376604, first differences of A054354.
For first differences we had A375735, ones A375713(n) - 1.
Positions of zeros are A376600, complement A376601.
A000961 lists prime-powers inclusive, exclusive A246655.
A007916 lists non-perfect-powers.
A057820 gives first differences of prime-powers inclusive, first appearances A376341, sorted A376340.
A321346/A321378 count integer partitions without prime-powers, factorizations A322452.
For non-prime-powers: A024619/A361102 (terms), A375735/A375708 (first differences), A376600 (inflections and undulations), A376601 (nonzero curvature).
For second differences: A036263 (prime), A073445 (composite), A376559 (perfect-power), A376562 (non-perfect-power), A376590 (squarefree), A376593 (nonsquarefree), A376596 (prime-power).
Cf. A025475, A053707, A064113, A093555, A174965, A251092, A333254, A376653, A376654.

Differences[Select[Range[100],!(#==1||PrimePowerQ[#])&],2]

from sympy import primepi, integer_nthroot
def A376599(n):
    def iterfun(f,n=0):
        m, k = n, f(n)
        while m != k: m, k = k, f(k)
        return m
    def f(x): return int(n+1+sum(primepi(integer_nthroot(x,k)[0]) for k in range(1,x.bit_length())))
    return (a:=iterfun(f,n))-((b:=iterfun(lambda x:f(x)+1,a))<<1)+iterfun(lambda x:f(x)+2,b) # Chai Wah Wu, Oct 02 2024

A078649 Numbers n such that A000002(n)=A000002(n+1) where A000002 is the Kolakoski sequence.

Original entry on oeis.org

2, 4, 8, 11, 13, 16, 18, 22, 26, 28, 31, 35, 38, 40, 44, 48, 51, 53, 56, 58, 62, 65, 67, 70, 74, 78, 80, 83, 85, 89, 92, 94, 97, 99, 103, 107, 110, 112, 115, 119, 121, 124, 126, 130, 133, 135, 138, 140, 144, 148, 150, 153, 157, 160, 162, 165, 167, 171, 175, 178, 180

Offset: 1

Benoit Cloitre, Dec 14 2002

Complement sequence of A054353. - Benoit Cloitre, Feb 07 2009
This sequence is the union of A074262 and A074263. - Nathaniel Johnston, May 02 2011
A054354(a(n)-1) = 0. - Reinhard Zumkeller, Aug 03 2013
This is a subsequence of A216345. In particular, this consists of A216345(i) such that A000002(i) = A216345(i+1)-A216345(i) = 2. A013948 is the sequence of all such i. - Danny Rorabaugh, Mar 13 2015

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

Cf. A054353, A074262, A074263, A013948, A216345.

a078649 n = a078649_list !! (n-1)
a078649_list = map (+ 1) $ filter ((== 0) . a054354) [1..]
-- Reinhard Zumkeller, Aug 03 2013

isA078649 := proc(n)
    if A000002(n) = A000002(n+1) then
        true;
    else
        false;
    end if;
end proc:
A078649 := proc(n)
    option remember;
    if n = 1 then
        2;
    else
        for a from procname(n-1)+1 do
            if isA078649(a) then
                return a;
            end if;
        end do:
    end if;
end proc:
seq(A078649(n),n=1..50) ; # R. J. Mathar, Nov 15 2014

a2 = {1, 2, 2}; Do[ a2 = Join[a2, {1+Mod[n-1, 2]}], {n, 3, 80}, {a2[[n]]}]; a3 = Accumulate[a2]; Complement[ Range[ Last[a3]], a3] (* Jean-François Alcover, Jun 18 2013 *)

a(n) is probably asymptotic to 3*n.

a(n) = A216345(A013948(n)). - Danny Rorabaugh, Mar 13 2015

cp's OEIS Frontend

A376604 Second differences of the Kolakoski sequence (A000002). First differences of A054354.

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A156728 a(n) = abs(A054354(n)).

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A000002 Kolakoski sequence: a(n) is length of n-th run; a(1) = 1; sequence consists just of 1's and 2's.

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A376593 Second differences of consecutive nonsquarefree numbers (A013929). First differences of A078147.

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A376596 Second differences of consecutive prime-powers inclusive (A000961). First differences of A057820.

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A376562 Second differences of consecutive non-perfect-powers (A007916). First differences of A375706.

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A376590 Second differences of consecutive squarefree numbers (A005117). First differences of A076259.

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A376559 Second differences of consecutive perfect powers (A001597). First differences of A053289.

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