A000452 -id:A000452 - cp's OEIS frontend

A031975 Erroneous version of A000452.

1, 2, 3, 5, 6, 7, 8, 10, 11, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 24, 26, 27, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 45, 47, 48, 50, 53

Offset: 1

dead

A174648 Partial sums of A000452.

Original entry on oeis.org

1, 3, 6, 11, 17, 24, 32, 42, 53, 66, 80, 95, 111, 128, 147, 168, 190, 213, 237, 263, 290, 319, 349, 380, 413, 447, 482, 519, 557, 596, 636, 677, 719, 762, 808, 855, 903, 954, 1007, 1061, 1116, 1172, 1229, 1287, 1346, 1407, 1469, 1534, 1600, 1667, 1736, 1806

Offset: 1

Jonathan Vos Post, Mar 25 2010

nonn

Partial sums of sequence starting with a(1) = 1 and for which a(n) is smallest number which avoids any 3-term G.P. (geometric progression). The subsequence of primes in this partial sum begins: 3, 11, 17, 53, 263, 349, 557, 677, 719, 1061, 1229, 1667, 1877, 2179, 3019, 3203. The subsequence of squares begins: 1, 1600. subsequence of powers of 2 begins: 1, 32, 128.

			a(9) = 1 + 2 + 3 + 5 + 6 + 7 + 8 + 10 + 11 = 53 is prime.

Cf. A000452.

a(n) = Sum_{i=1..n} A000452(i).

A005836 Numbers whose base-3 representation contains no 2.

Original entry on oeis.org

0, 1, 3, 4, 9, 10, 12, 13, 27, 28, 30, 31, 36, 37, 39, 40, 81, 82, 84, 85, 90, 91, 93, 94, 108, 109, 111, 112, 117, 118, 120, 121, 243, 244, 246, 247, 252, 253, 255, 256, 270, 271, 273, 274, 279, 280, 282, 283, 324, 325, 327, 328, 333, 334, 336, 337, 351, 352

Offset: 1

N. J. A. Sloane, Jeffrey Shallit

3 does not divide binomial(2s, s) if and only if s is a member of this sequence, where binomial(2s, s) = A000984(s) are the central binomial coefficients.
This is the lexicographically earliest increasing sequence of nonnegative numbers that contains no arithmetic progression of length 3. - Robert Craigen (craigenr(AT)cc.umanitoba.ca), Jan 29 2001
In the notation of A185256 this is the Stanley Sequence S(0,1). - N. J. A. Sloane, Mar 19 2010
Complement of A074940. - Reinhard Zumkeller, Mar 23 2003
Sums of distinct powers of 3. - Ralf Stephan, Apr 27 2003
Numbers n such that central trinomial coefficient A002426(n) == 1 (mod 3). - Emeric Deutsch and Bruce E. Sagan, Dec 04 2003
A039966(a(n)+1) = 1; A104406(n) = number of terms <= n.
Subsequence of A125292; A125291(a(n)) = 1 for n>1. - Reinhard Zumkeller, Nov 26 2006
Also final value of n - 1 written in base 2 and then read in base 3 and with finally the result translated in base 10. - Philippe LALLOUET (philip.lallouet(AT)wanadoo.fr), Jun 23 2007
a(n) modulo 2 is the Thue-Morse sequence A010060. - Dennis Tseng, Jul 16 2009
Also numbers such that the balanced ternary representation is the same as the base 3 representation. - Alonso del Arte, Feb 25 2011
Fixed point of the morphism: 0 -> 01; 1 -> 34; 2 -> 67; ...; n -> (3n)(3n+1), starting from a(1) = 0. - Philippe Deléham, Oct 22 2011
It appears that this sequence lists the values of n which satisfy the condition sum(binomial(n, k)^(2*j), k = 0..n) mod 3 <> 0, for any j, with offset 0. See Maple code. - Gary Detlefs, Nov 28 2011
Also, it follows from the above comment by Philippe Lallouet that the sequence must be generated by the rules: a(1) = 0, and if m is in the sequence then so are 3*m and 3*m + 1. - L. Edson Jeffery, Nov 20 2015
Add 1 to each term and we get A003278. - N. J. A. Sloane, Dec 01 2019

			12 is a term because 12 = 110_3.
This sequence regarded as a triangle with rows of lengths 1, 1, 2, 4, 8, 16, ...:
   0
   1
   3,  4
   9, 10, 12, 13
  27, 28, 30, 31, 36, 37, 39, 40
  81, 82, 84, 85, 90, 91, 93, 94, 108, 109, 111, 112, 117, 118, 120, 121
... - _Philippe Deléham_, Jun 06 2015

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section E10, pp. 317-323.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

David W. Wilson, Table of n, a(n) for n = 1..10000 (first 1024 terms from T. D. Noe)
J.-P. Allouche, G.-N. Han, and Jeffrey Shallit, On some conjectures of P. Barry, arXiv:2006.08909 [math.NT], 2020.
J.-P. Allouche and Jeffrey Shallit, The ring of k-regular sequences, Theoretical Computer Sci., 98 (1992), 163-197.
J.-P. Allouche and Jeffrey Shallit, The ring of k-regular sequences, Theoretical Computer Sci., 98 (1992), 163-197.
J.-P. Allouche, Jeffrey Shallit and G. Skordev, Self-generating sets, integers with missing blocks and substitutions, Discrete Math. 292 (2005) 1-15.
David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.]
Megumi Asada, Bruce Fang, Eva Fourakis, Sarah Manski, Nathan McNew, Steven J. Miller, Gwyneth Moreland, Ajmain Yamin, and Sindy Xin Zhang, Avoiding 3-Term Geometric Progressions in Hurwitz Quaternions, Williams College (2023).
Robert Baillie and Thomas Schmelzer, Summing Kempner's Curious (Slowly-Convergent) Series, Mathematica Notebook kempnerSums.nb, Wolfram Library Archive, 2008.
Noam Benson-Tilsen, Samuel Brock, Brandon Faunce, Monish Kumar, Noah Dokko Stein, and Joshua Zelinsky, Total Difference Labeling of Regular Infinite Graphs, arXiv:2107.11706 [math.CO], 2021.
Raghavendra Bhat, Cristian Cobeli, and Alexandru Zaharescu, Filtered rays over iterated absolute differences on layers of integers, arXiv:2309.03922 [math.NT], 2023. See page 16.
Matvey Borodin, Hannah Han, Kaylee Ji, Tanya Khovanova, Alexander Peng, David Sun, Isabel Tu, Jason Yang, William Yang, Kevin Zhang, and Kevin Zhao, Variants of Base 3 over 2, arXiv:1901.09818 [math.NT], 2019.
Ben Chen, Richard Chen, Joshua Guo, Tanya Khovanova, Shane Lee, Neil Malur, Nastia Polina, Poonam Sahoo, Anuj Sakarda, Nathan Sheffield, and Armaan Tipirneni, On Base 3/2 and its Sequences, arXiv:1808.04304 [math.NT], 2018.
Karl Dilcher and Larry Ericksen, Hyperbinary expansions and Stern polynomials, Elec. J. Combin, Vol. 22 (2015), Article P2.24.
P. Erdős, V. Lev, G. Rauzy, C. Sandor, and A. Sarkozy, Greedy algorithm, arithmetic progressions, subset sums and divisibility, Discrete Math., Vol. 200, No. 1-3 (1999), pp. 119-135 (see Table 1). alternate link.
Joseph L. Gerver and L. Thomas Ramsey, Sets of integers with no long arithmetic progressions generated by the greedy algorithm, Math. Comp., Vol. 33, No. 148 (1979), pp. 1353-1359.
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, p. 45.
Ryota Inagaki, Tanya Khovanova, and Austin Luo, Permutation-based Strategies for Labeled Chip-Firing on k-ary Trees, arXiv:2503.09577 [math.CO], 2025. See p. 18.
Kathrin Kostorz, Robert W. Hölzel and Katharina Krischer, Distributed coupling complexity in a weakly coupled oscillatory network with associative properties, New J. Phys., Vol. 15 (2013), #083010; doi:10.1088/1367-2630/15/8/083010.
Clark Kimberling, Affinely recursive sets and orderings of languages, Discrete Math., Vol. 274, Vol. 1-3 (2004), pp. 147-160.
John W. Layman, Some Properties of a Certain Nonaveraging Sequence, J. Integer Sequences, Vol. 2 (1999), Article 99.1.3.
Manfred Madritsch and Stephan Wagner, A central limit theorem for integer partitions, Monatsh. Math., Vol. 161, No. 1 (2010), pp. 85-114.
Richard A. Moy and David Rolnick, Novel structures in Stanley sequences, Discrete Math., Vol. 339, No. 2 (2016), pp. 689-698; arXiv preprint, arXiv:1502.06013 [math.CO], 2015.
A. M. Odlyzko and R. P. Stanley, Some curious sequences constructed with the greedy algorithm, 1978, remark 1 (PDF, PS, TeX).
Paul Pollack, Analytic and Combinatorial Number Theory Course Notes, p. 228. [?Broken link]
Paul Pollack, Analytic and Combinatorial Number Theory Course Notes, p. 228.
David Rolnick and Praveen S. Venkataramana, On the growth of Stanley sequences, Discrete Math., Vol. 338, No. 11 (2015), pp. 1928-1937, see p. 1930; arXiv preprint, arXiv:1408.4710 [math.CO], 2014.
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS.
Ralf Stephan, Divide-and-conquer generating functions. I. Elementary sequences, arXiv:math/0307027 [math.CO], 2003.
Ralf Stephan, Some divide-and-conquer sequences with (relatively) simple ordinary generating functions, 2004.
Ralf Stephan, Table of generating functions.
Zoran Sunic, Tree morphisms, transducers and integer sequences, arXiv:math/0612080 [math.CO], 2006.
B. Vasic, K. Pedagani and M. Ivkovic, High-rate girth-eight low-density parity-check codes on rectangular integer lattices, IEEE Transactions on Communications, Vol. 52, Issue 8 (2004), pp. 1248-1252.
Eric Weisstein's World of Mathematics, Central Binomial Coefficient.
Index entries for 3-automatic sequences.

Cf. A039966 (characteristic function).
Cf. A002426, A004793, A005823, A007088, A007089, A032924, A033042-A033052, A054591, A055246, A062548, A065361, A074940, A081601, A081603, A081611, A083096, A089118, A121153, A170943, A185256.
For generating functions Product_{k>=0} (1+a*x^(b^k)) for the following values of (a,b) see: (1,2) A000012 and A000027, (1,3) A039966 and A005836, (1,4) A151666 and A000695, (1,5) A151667 and A033042, (2,2) A001316, (2,3) A151668, (2,4) A151669, (2,5) A151670, (3,2) A048883, (3,3) A117940, (3,4) A151665, (3,5) A151671, (4,2) A102376, (4,3) A151672, (4,4) A151673, (4,5) A151674.
Row 3 of array A104257.
Summary of increasing sequences avoiding arithmetic progressions of specified lengths (the second of each pair is obtained by adding 1 to the first):
   3-term AP: A005836 (>=0), A003278 (>0);
   4-term AP: A005839 (>=0), A005837 (>0);
   5-term AP: A020654 (>=0), A020655 (>0);
   6-term AP: A020656 (>=0), A005838 (>0);
   7-term AP: A020657 (>=0), A020658 (>0);
   8-term AP: A020659 (>=0), A020660 (>0);
   9-term AP: A020661 (>=0), A020662 (>0);
  10-term AP: A020663 (>=0), A020664 (>0).
See also A000452.

a005836 n = a005836_list !! (n-1)
a005836_list = filter ((== 1) . a039966) [0..]
-- Reinhard Zumkeller, Jun 09 2012, Sep 29 2011

function a(n)
    m, r, b = n, 0, 1
    while m > 0
        m, q = divrem(m, 2)
        r += b * q
        b *= 3
    end
r end; [a(n) for n in 0:57] |> println # Peter Luschny, Jan 03 2021

t := (j, n) -> add(binomial(n,k)^j, k=0..n):
for i from 1 to 400 do
    if(t(4,i) mod 3 <>0) then print(i) fi
od; # Gary Detlefs, Nov 28 2011
# alternative Maple program:
a:= proc(n) option remember: local k, m:
if n=1 then 0 elif n=2 then 1 elif n>2 then k:=floor(log[2](n-1)): m:=n-2^k: procname(m)+3^k: fi: end proc:
seq(a(n), n=1.. 20); # Paul Weisenhorn, Mar 22 2020
# third Maple program:
a:= n-> `if`(n=1, 0, irem(n-1, 2, 'q')+3*a(q+1)):
seq(a(n), n=1..100);  # Alois P. Heinz, Jan 26 2022

Table[FromDigits[IntegerDigits[k, 2], 3], {k, 60}]
Select[Range[0, 400], DigitCount[#, 3, 2] == 0 &] (* Harvey P. Dale, Jan 04 2012 *)
Join[{0}, Accumulate[Table[(3^IntegerExponent[n, 2] + 1)/2, {n, 57}]]] (* IWABUCHI Yu(u)ki, Aug 01 2012 *)
FromDigits[#,3]&/@Tuples[{0,1},7] (* Harvey P. Dale, May 10 2019 *)

A=vector(100);for(n=2,#A,A[n]=if(n%2,3*A[n\2+1],A[n-1]+1));A \\ Charles R Greathouse IV, Jul 24 2012

is(n)=while(n,if(n%3>1,return(0));n\=3);1 \\ Charles R Greathouse IV, Mar 07 2013

a(n) = fromdigits(binary(n-1),3);  \\ Gheorghe Coserea, Jun 15 2018

def A005836(n):
    return int(format(n-1,'b'),3) # Chai Wah Wu, Jan 04 2015

a(n) = A005823(n)/2 = A003278(n)-1 = A033159(n)-2 = A033162(n)-3.
Numbers n such that the coefficient of x^n is > 0 in prod (k >= 0, 1 + x^(3^k)). - Benoit Cloitre, Jul 29 2003
a(n+1) = Sum_{k=0..m} b(k)* 3^k and n = Sum( b(k)* 2^k ).
a(2n+1) = 3a(n+1), a(2n+2) = a(2n+1) + 1, a(0) = 0.
a(n+1) = 3*a(floor(n/2)) + n - 2*floor(n/2). - Ralf Stephan, Apr 27 2003
G.f.: (x/(1-x)) * Sum_{k>=0} 3^k*x^2^k/(1+x^2^k). - Ralf Stephan, Apr 27 2003
a(n) = Sum_{k = 1..n-1} (1 + 3^A007814(k)) / 2. - Philippe Deléham, Jul 09 2005
From Reinhard Zumkeller, Mar 02 2008: (Start)
A081603(a(n)) = 0.
If the offset were changed to zero, then: a(0) = 0, a(n+1) = f(a(n)+1, a(n)+1) where f(x, y) = if x < 3 and x <> 2 then y else if x mod 3 = 2 then f(y+1, y+1) else f(floor(x/3), y). (End)
With offset a(0) = 0: a(n) = Sum_{k>=0} A030308(n,k)*3^k. - Philippe Deléham, Oct 15 2011
a(2^n) = A003462(n). - Philippe Deléham, Jun 06 2015
We have liminf_{n->infinity} a(n)/n^(log(3)/log(2)) = 1/2 and limsup_{n->infinity} a(n)/n^(log(3)/log(2)) = 1. - Gheorghe Coserea, Sep 13 2015
a(2^k+m) = a(m) + 3^k with 1 <= m <= 2^k and 1 <= k, a(1)=0, a(2)=1. - Paul Weisenhorn, Mar 22 2020
Sum_{n>=2} 1/a(n) = 2.682853110966175430853916904584699374821677091415714815171756609672281184705... (calculated using Baillie and Schmelzer's kempnerSums.nb, see Links). - Amiram Eldar, Feb 12 2022
A065361(a(n)) = n-1. - Rémy Sigrist, Feb 06 2023
a(n) ≍ n^k, where k = log 3/log 2 = 1.5849625007. (I believe the constant varies from 1/2 to 1.) - Charles R Greathouse IV, Mar 29 2024

Offset corrected by N. J. A. Sloane, Mar 02 2008

Edited by the Associate Editors of the OEIS, Apr 07 2009

A224853 Lexicographically earliest sequence of nonnegative integers which does not contain a three-term arithmetic, geometric, or harmonic subsequence.

Original entry on oeis.org

0, 1, 3, 4, 10, 11, 13, 14, 29, 30, 32, 33, 38, 39, 41, 42, 85, 86, 88, 89, 94, 95, 97, 98, 112, 113, 115, 116, 122, 123, 125, 238, 248, 251, 252, 255, 257, 260, 261, 273, 275, 278, 279, 287, 288, 292, 330, 331, 334, 335

Offset: 1

Timur Vural, Jul 28 2013

This sequence diverges from A225571 at 477th term. Here a(477) = 17408, while A225571(477) = 17380. - Giovanni Resta, Jul 29 2013

			After terms 0, 1, 3, 4 have been added, the terms 5,...,9 are forbidden by subsequences (3,4,5), (0,3,6), (1,4,7), (0,4,8) and (1,3,9) so the next term is 10.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A225571, A003278, A005836, A000452.

# Program that generates all values of a(x) less than a given input n.
def a(n):
      seq=[0, 1]
      for x in range(2, n+1):
          c=0
          for y in seq:
              if (x+y)/2 not in seq:
                  if (x*y)**0.5 not in seq[1:]:
                      if (2*x*y)/(x+y) not in seq[1:]:
                          c+=1
          if c==len(seq):
              seq.append(x)
      return seq

A225571 Lexicographically earliest sequence of nonnegative integers which does not contain a three-term arithmetic or geometric subsequence.

Original entry on oeis.org

Offset: 1

Giovanni Resta, Jul 29 2013

This sequence diverges from A224853 at 477th term. Here a(477) = 17380, while A224853(477) = 17408.

			After terms 0, 1, 3, 4 have been added, the terms 5,...,9 are forbidden by subsequences (3,4,5), (0,3,6), (1,4,7), (0,4,8) and (1,3,9) so the next term is 10.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A224853, A003278, A005836, A000452.

seq = {0, 1}; bad[n_] := Catch[ Do[If[MemberQ[seq, (n + e)/2], Throw@True], {e, seq}];  Do[If[MemberQ[seq, Sqrt[n*e]], Throw@True], {e, Rest@ seq}]; False]; While[Length[seq] < 100, x = Last[seq]+1; While[bad[x], x++]; AppendTo[seq, x]]; seq

A268811 Sequence of positive integers where each is chosen to be as small as possible subject to the condition that no three terms a(j), a(j+k), a(j+2k) (for any j and k) form a geometric progression.

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 2, 3, 3, 1, 1, 2, 1, 1, 2, 2, 3, 3, 2, 3, 3, 5, 5, 6, 5, 5, 6, 1, 1, 2, 1, 1, 2, 2, 3, 3, 1, 1, 2, 1, 1, 2, 2, 3, 3, 2, 3, 3, 5, 5, 6, 5, 5, 6, 2, 3, 3, 5, 5, 6, 5, 5, 6, 6, 7, 7, 6, 7, 7, 8, 8, 10, 6, 7, 7, 6, 7, 7, 8, 8, 10, 1, 1, 2, 1, 1, 2, 2, 3, 3, 1, 1, 2, 1, 1, 2, 2, 3, 3, 2, 3, 3, 5, 5, 6, 5, 5, 6, 1, 1, 2, 1, 1, 2, 2, 3, 3, 1, 1, 2, 1, 1, 2, 2, 3, 3, 2

Offset: 1

Aaron David Fairbanks, Feb 13 2016

Apparently: all terms belong to A000452, and for any k > 0, the value A000452(k) first appears at index A265316(k+1). - Rémy Sigrist, May 13 2021

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, C program for A268811

Cf. A000452, A229037, A265316.

C
```
// See Links section.
```

A268811_list = []
for n in range(1000):
    i, j, b = 1, 1, set()
    while n-2*i >= 0:
        b.add(A268811_list[n-i]**2/A268811_list[n-2*i])
        i += 1
        while j in b:
            b.remove(j)
            j += 1
    A268811_list.append(j)

A289206 Greedy strictly increasing sequence starting at a(1)=1 avoiding both arithmetic and geometric progressions of length 3.

Original entry on oeis.org

1, 2, 5, 6, 12, 13, 15, 16, 32, 33, 35, 39, 40, 42, 56, 81, 84, 85, 88, 90, 93, 94, 108, 109, 113, 115, 116, 159, 189, 207, 208, 222, 223, 232, 235, 240, 243, 244, 249, 250, 252, 259, 267, 271, 289, 304, 314, 318, 325, 340, 342, 397, 504, 508, 511, 531, 549

Offset: 1

Roderick MacPhee, Jun 28 2017

nonn

By avoiding arithmetic progressions, at most 2/3 of the numbers up to a(n) are in the sequence. The sequence doesn't contain 3 consecutive powers in arithmetic progression for any base c.

Where a(n)+1 = a(n+1): 1, 3, 5, 7, 9, 12, 17, 21, 23, 26, 30, 32, 37, 39, etc. - Robert G. Wilson v, Jul 02 2017

			5 is in the sequence because 1,2,5 is neither an arithmetic progression nor a geometric progression.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Math StackExchange, A sequence that avoids both arithmetic and geometric progression (2014)

Cf. A000452, A003278, A005836, A224853, A225571.

{my(a=[1,2]);
for(x=3,100,
if(#select(r->#select(q->q==2*r,b)==0,b=vecsort(apply(r->x-r,a)))==#a && #select(r->#select(q->q==r^2,b)==0,b=vecsort(apply(r->x/r,a)))==#a,a=concat(a,x)));a
}

first(n)=my(v=vector(n)); v[1]=1; for(k=2,n, my(avoid=List(),t,last=v[k-1]); for(i=2,k-1, for(j=1,i-1, t=2*v[i]-v[j]; if(t>last, listput(avoid, t)); if(denominator(t=v[i]^2/v[j])==1 && t>last, listput(avoid,t)))); avoid=Set(avoid); for(i=v[k-1]+1,v[k-1]+#avoid+1, if(!setsearch(avoid,i), v[k]=i; break))); v \\ Charles R Greathouse IV, Jun 29 2017

a(n) >= 3n/2 for n > 2.

More terms from Alois P. Heinz, Jun 28 2017

A381137 Lexicographically earliest sequence of distinct positive integers such that no 3 terms are in harmonic progression.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 43, 44, 46, 47, 48, 49, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 73, 74, 76, 79, 81, 82, 83, 85, 86

Offset: 1

Neal Gersh Tolunsky, Feb 15 2025

nonn

A harmonic progression is a sequence of values whose reciprocals are in arithmetic progression. Equivalently, if (a, b, c) is a harmonic progression, then b is the harmonic mean of a and c.
a(n) is the smallest integer greater than a(n-1) which does not form a 3-term harmonic progression with 2 previously occurring terms.
Every prime occurs in the sequence.

			6 is not a term in the sequence because it would form a harmonic progression with 2 and 3, which occurred earlier. The progression (1/6, 1/3, 1/2) has common difference 1/6.

Neal Gersh Tolunsky, Table of n, a(n) for n = 1..10000

Analogous sequences: A003278 (for arithmetic progressions), A000452 (for geometric progressions).

Cf. A005279, A174905.

from itertools import count
def A381137_generator():
    a_list = []
    forbidden = set()
    a = 0
    while 1:
        a = next(k for k in count(a+1) if k not in forbidden)
        yield a
        forbidden.update(a*b//m for b in a_list if (m:=2*b-a) > 0 and a*b%m == 0)
        a_list.append(a) # Pontus von Brömssen, Mar 04 2025

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A031975 Erroneous version of A000452.

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A174648 Partial sums of A000452.

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A005836 Numbers whose base-3 representation contains no 2.

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A224853 Lexicographically earliest sequence of nonnegative integers which does not contain a three-term arithmetic, geometric, or harmonic subsequence.

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A225571 Lexicographically earliest sequence of nonnegative integers which does not contain a three-term arithmetic or geometric subsequence.

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A268811 Sequence of positive integers where each is chosen to be as small as possible subject to the condition that no three terms a(j), a(j+k), a(j+2k) (for any j and k) form a geometric progression.

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A289206 Greedy strictly increasing sequence starting at a(1)=1 avoiding both arithmetic and geometric progressions of length 3.

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A381137 Lexicographically earliest sequence of distinct positive integers such that no 3 terms are in harmonic progression.

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