A005836 -id:A005836 - cp's OEIS frontend

A235266 Primes whose base-2 representation is also the base-3 representation of a prime.

2, 7, 11, 13, 41, 47, 67, 73, 79, 109, 127, 151, 173, 181, 191, 193, 211, 223, 227, 229, 233, 251, 283, 331, 367, 421, 443, 487, 541, 557, 563, 587, 601, 607, 631, 641, 661, 677, 719, 733, 877, 941, 947, 967, 971, 1033, 1187, 1193, 1201, 1301, 1321, 1373, 1447, 1451, 1471, 1531, 1567, 1571, 1657, 1667, 1669, 1697, 1709, 1759

Offset: 1

M. F. Hasler, Jan 05 2014

Robert Israel, Table of n, a(n) for n = 1..10000
M. F. Hasler, Primes whose base c expansion is also the base b expansion of a prime

Cf. A090707 - A091924, A235461 - A235482. See the LINK for further cross-references.

Cf. A077717, A005836.

f:= proc(n) local L,i;
  L:= convert(n,base,2);
  isprime(add(L[i]*3^(i-1),i=1..nops(L)))
end proc:
select(f, [seq(ithprime(i),i=1..1000)]); # Robert Israel, Jun 03 2019

Select[Prime@ Range@ 250, PrimeQ@ FromDigits[IntegerDigits[#, 2], 3] &] (* Michael De Vlieger, Jun 03 2019 *)

is(p,b=3,c=2)=isprime(vector(#d=digits(p,c),i,b^(#d-i))*d~)&&isprime(p) \\ This code can be used for other bases b,c when b>c. See A235265 for code valid for b

forprime(p=2, 1e3, if(isprime(fromdigits(binary(p), 3)), print1(p", "))) \\ Charles R Greathouse IV, Mar 28 2022

from sympy import isprime, nextprime
def agen(): # generator of terms
    p = 2
    while True:
        p3 = sum(3**i for i, bi in enumerate(bin(p)[2:][::-1]) if bi=='1')
        if isprime(p3):
            yield p
        p = nextprime(p)
g = agen()
print([next(g) for n in range(1, 65)]) # Michael S. Branicky, Jan 16 2022

a(n) is the number whose base-3 representation is the base-2 representation of A235265(n).

A005823 Numbers whose ternary expansion contains no 1's.

Original entry on oeis.org

0, 2, 6, 8, 18, 20, 24, 26, 54, 56, 60, 62, 72, 74, 78, 80, 162, 164, 168, 170, 180, 182, 186, 188, 216, 218, 222, 224, 234, 236, 240, 242, 486, 488, 492, 494, 504, 506, 510, 512, 540, 542, 546, 548, 558, 560, 564, 566, 648, 650, 654, 656, 666, 668, 672, 674

Offset: 1

N. J. A. Sloane, Jeffrey Shallit

The set of real numbers between 0 and 1 that contain no 1's in their ternary expansion is the well-known Cantor set with Hausdorff dimension log 2 / log 3.
Complement of A081606. - Reinhard Zumkeller, Mar 23 2003
Numbers k such that the k-th Apery number is congruent to 1 (mod 3) (cf. A005258). - Benoit Cloitre, Nov 30 2003
Numbers k such that the k-th central Delannoy number is congruent to 1 (mod 3) (cf. A001850). - Benoit Cloitre, Nov 30 2003
Numbers k such that there exists a permutation p_1, ..., p_k of 1, ..., k such that i + p_i is a power of 3 for every i. - Ray Chandler, Aug 03 2004
Subsequence of A125292. - Reinhard Zumkeller, Nov 26 2006
The first 2^n terms of the sequence could be obtained using the Cantor process for the segment [0,3^n-1]. E.g., for n=2 we have [0,{1},2,{3,4,5},6,{7},8]. The numbers outside of braces are the first 4 terms of the sequence. Therefore the terms of the sequence could be called "Cantor's numbers". - Vladimir Shevelev, Jun 13 2008
Mahler proved that positive a(n) is never a square. - Michel Marcus, Nov 12 2012
Define t: Z -> P(R) so that t(k) is the translated Cantor ternary set spanning [k, k+1], and let T be the union of t(a(n)) for all n. T = T * 3 = T / 3 is the closure of the Cantor ternary set under multiplication by 3. - Peter Munn, Oct 30 2019

K. J. Falconer, The Geometry of Fractal Sets, Cambridge, 1985; p. 14.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Winston de Greef, Table of n, a(n) for n = 1..16384 (first 1024 terms from T. D. Noe)
J.-P. Allouche and Jeffrey Shallit, The ring of k-regular sequences, Theoretical Computer Sci., Vol. 98, No. 2 (1992), pp. 163-197.
J.-P. Allouche and Jeffrey Shallit, The ring of k-regular sequences, Theoretical Computer Sci., Vol. 98, No. 2 (1992), pp. 163-197.
Robert Baillie and Thomas Schmelzer, Summing Kempner's Curious (Slowly-Convergent) Series, Mathematica Notebook kempnerSums.nb, Wolfram Library Archive, 2008.
Sajed Haque, Chapter 3.4 of Discriminators of Integer Sequences, 2017, See p. 45.
Sajed Haque and Jeffrey Shallit, Discriminators and k-Regular Sequences, arXiv:1605.00092 [cs.DM], 2016.
Clark Kimberling, Affinely recursive sets and orderings of languages, Discrete Math., Vol. 274, No. 1-3 (2004), pp. 147-160.
Kurt Mahler, The representation of squares to the base 3, Acta Arith., Vol. 53, Issue 1 (1989), pp. 99-106.
M. Mendes France and A. J. van der Poorten, From geometry to Euler identities, Theoret. Comput. Sci., Vol. 65, No. 2 (1989), pp. 213-220.
Eric Weisstein's World of Mathematics, Cantor Set.
Index entries for 3-automatic sequences.

Twice A005836.
Cf. A032924, A014263, A007089, A062756, A061392, A001196, A097252-A097262.
Cf. A088917 (characteristic function), A306556.

a:= proc(n) option remember;
      `if`(n=1, 0, `if`(irem (n, 2, 'q')=0, 3*a(q)+2, 3*a(q+1)))
    end:
seq(a(n), n=1..100); # Alois P. Heinz, Apr 19 2012

Select[ Range[ 0, 729 ], (Count[ IntegerDigits[ #, 3 ], 1 ]==0)& ]
Select[Range[0,700],DigitCount[#,3,1]==0&] (* Harvey P. Dale, Mar 12 2016 *)

is(n)=while(n,if(n%3==1,return(0),n\=3));1 \\ Charles R Greathouse IV, Apr 20 2012

a(n)=n=binary(n-1);sum(i=1,#n,2*n[i]*3^(#n-i)) \\ Charles R Greathouse IV, Apr 20 2012

a(n)=2*fromdigits(binary(n-1),3) \\ Charles R Greathouse IV, Aug 24 2016

def A005823(n):
    return 2*int(format(n-1,'b'),3) # Chai Wah Wu, Jan 04 2015

a(n) = 2 * A005836(n).
a(2n) = 3*a(n)+2, a(2n+1) = 3*a(n+1), a(1) = 0.
a(n) = Sum_{k = 1..n} 1 + 3^A007814(k). - Philippe Deléham, Jul 09 2005
A125291(a(n)) = 1 for n>0. - Reinhard Zumkeller, Nov 26 2006
From Reinhard Zumkeller, Mar 02 2008: (Start)
A062756(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 <> 1 then y else if x mod 3 = 1 then f(y+1, y+1) else f(floor(x/3), y). (End)
G.f. g(x) satisfies g(x) = 3*g(x^2)*(1+1/x) + 2*x^2/(1-x^2). - Robert Israel, Jan 04 2015
Sum_{n>=2} 1/a(n) = 1.341426555483087715426958452292349687410838545707857407585878304836140592352... (calculated using Baillie and Schmelzer's kempnerSums.nb, see Links). - Amiram Eldar, Feb 12 2022

More terms from Sascha Kurz, Mar 24 2002

Offset corrected by N. J. A. Sloane, Mar 02 2008. This may require some of the formulas to be adjusted.

A011539 "9ish numbers": decimal representation contains at least one nine.

Original entry on oeis.org

9, 19, 29, 39, 49, 59, 69, 79, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 109, 119, 129, 139, 149, 159, 169, 179, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 209, 219, 229, 239, 249, 259, 269, 279, 289, 290, 291, 292, 293, 294, 295, 296, 297, 298

Offset: 1

N. J. A. Sloane

The 9ish numbers are closed under lunar multiplication. The lunar primes (A087097) are a subset.
Almost all numbers are 9ish, in the sense that the asymptotic density of this set is 1: Among the 9*10^(n-1) n-digit numbers, only a fraction of 0.8*0.9^(n-1) doesn't have a digit 9, and this fraction tends to zero (< 1/10^k for n > 22k-3). This explains the formula a(n) ~ n. - M. F. Hasler, Nov 19 2018
A 9ish number is a number whose largest decimal digit is 9. - Stefano Spezia, Nov 16 2023

			E.g. 9, 19, 69, 90, 96, 99 and 1234567890 are all 9ish.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
D. Applegate, C program for lunar arithmetic and number theory [Note: we have now changed the name from "dismal arithmetic" to "lunar arithmetic" - the old name was too depressing]
D. Applegate, M. LeBrun, and N. J. A. Sloane, Dismal Arithmetic, arXiv:1107.1130 [math.NT], 2011. [Note: we have now changed the name from "dismal arithmetic" to "lunar arithmetic" - the old name was too depressing]
Index entries for sequences related to dismal (or lunar) arithmetic
Index entries for 10-automatic sequences.

Cf. A088924 (number of n-digit terms).
Cf. A087062 (lunar product), A087097 (lunar primes).
A102683 (number of digits 9 in n); fixed points > 8 of A068505.
Cf. Numbers with at least one digit b-1 in base b : A074940 (b=3), A337250 (b=4), A337572 (b=5), A333656 (b=6), A337141 (b=7), A337239 (b=8), A338090 (b=9), this sequence (b=10), A095778 (b=11).
Cf. Numbers with no digit b-1 in base b: A005836 (b=3), A023717 (b=4), A020654 (b=5), A037465 (b=6), A020657 (b=7), A037474 (b=8), A037477 (b=9), A007095 (b=10), A171397 (b=11).
Supersequence of A043525.

Filtered([1..300],n->9 in ListOfDigits(n)); # Muniru A Asiru, Feb 25 2019

a011539 n = a011539_list !! (n-1)
a011539_list = filter ((> 0) . a102683) [1..]  -- Reinhard Zumkeller, Dec 29 2011

seq(`if`(numboccur(9, convert(n, base, 10))>0, n, NULL), n=0..100); # François Marques, Oct 12 2020

Select[ Range[ 0, 100 ], (Count[ IntegerDigits[ #, 10 ], 9 ]>0)& ] (* François Marques, Oct 12 2020 *)
Select[Range[300],DigitCount[#,10,9]>0&] (* Harvey P. Dale, Mar 04 2023 *)

is(n)=n=vecsort(digits(n));n[#n]==9 \\ Charles R Greathouse IV, May 15 2013

select( is_A011539(n)=vecmax(digits(n))==9, [1..300]) \\ M. F. Hasler, Nov 16 2018

def ok(n): return '9' in str(n)
print(list(filter(ok, range(299)))) # Michael S. Branicky, Sep 19 2021

def A011539(n):
    def f(x):
        l = (s:=str(x)).find('9')
        if l >= 0: s = s[:l]+'8'*(len(s)-l)
        return n+int(s,9)
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Dec 04 2024

Complement of A007095. A102683(a(n)) > 0 (defines this sequence). A068505(a(n)) = a(n): fixed points of A068505 are the terms of this sequence and the numbers < 9. - Reinhard Zumkeller, Dec 29 2011, edited by M. F. Hasler, Nov 16 2018

a(n) ~ n. - Charles R Greathouse IV, May 15 2013

A048883 a(n) = 3^wt(n), where wt(n) = A000120(n).

Original entry on oeis.org

1, 3, 3, 9, 3, 9, 9, 27, 3, 9, 9, 27, 9, 27, 27, 81, 3, 9, 9, 27, 9, 27, 27, 81, 9, 27, 27, 81, 27, 81, 81, 243, 3, 9, 9, 27, 9, 27, 27, 81, 9, 27, 27, 81, 27, 81, 81, 243, 9, 27, 27, 81, 27, 81, 81, 243, 27, 81, 81, 243, 81, 243, 243, 729, 3, 9, 9, 27, 9, 27, 27, 81, 9, 27, 27, 81, 27, 81

Offset: 0

John W. Layman

Or, a(n)=number of 1's ("live" cells) at stage n of a 2-dimensional cellular automata evolving by the rule: 1 if NE+NW+S=1, else 0.
This is the odd-rule cellular automaton defined by OddRule 013 (see Ekhad-Sloane-Zeilberger "Odd-Rule Cellular Automata on the Square Grid" link). - N. J. A. Sloane, Feb 25 2015
Or, start with S=[1]; replace S by [S, 3*S]; repeat ad infinitum.
Fixed point of the morphism 1 -> 13, 3 -> 39, 9 -> 9(27), ... = 3^k -> 3^k 3^(k+1), ... starting from a(0) = 1; 1 -> 13 -> 1339 -> = 1339399(27) -> 1339399(27)399(27)9(27)(27)(81) -> ..., . - Robert G. Wilson v, Jan 24 2006
Equals row sums of triangle A166453 (the square of Sierpiński's gasket, A047999). - Gary W. Adamson, Oct 13 2009
First bisection of A169697=1,5,3,19,3,. a(2n+2)+a(2n+3)=12,12,36,=12*A147610 ? Distribution of terms (in A000244): A011782=1,A000079 for first array, A000079 for second. - Paul Curtz, Apr 20 2010
a(A000225(n)) = A000244(n) and a(m) != A000244(n) for m < A000225(n). - Reinhard Zumkeller, Nov 14 2011
This sequence pertains to phenotype Punnett square mathematics. Start with X=1. Each hybrid cross involves the equation X:3X. Therefore, the ratio in the first (mono) hybrid cross is X=1:3X=3(1) or 3; or 3:1. When you move up to the next hybridization level, replace the previous cross ratio with X. X now represents 2 numbers-1:3. Therefore, the ratio in the second (di) hybrid cross is X=(1:3):3X=[3(1):3(3)] or (3:9). Put it together and you get 1:3:3:9. Each time you move up a hybridization level, replace the previous ratio with X, and use the same equation-X:3X to get its ratio. - John Michael Feuk, Dec 10 2011
Number of odd values in the n-th layer of Pascal's tetrahedron (see A268240). - Caden Le, Mar 03 2025
a(x*y) <= a(x)^A000120(y). - Joe Amos, Mar 28 2025

			From _Omar E. Pol_, Jun 07 2009: (Start)
Triangle begins:
  1;
  3;
  3,9;
  3,9,9,27;
  3,9,9,27,9,27,27,81;
  3,9,9,27,9,27,27,81,9,27,27,81,27,81,81,243;
  3,9,9,27,9,27,27,81,9,27,27,81,27,81,81,243,9,27,27,81,27,81,81,243,27,...
Or
  1;
  3,3;
  9,3,9,9;
  27,3,9,9,27,9,27,27;
  81,3,9,9,27,9,27,27,81,9,27,27,81,27,81,81;
  243,3,9,9,27,9,27,27,81,9,27,27,81,27,81,81,243,9,27,27,81,27,81,81,243,27...
(End)

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
David Applegate, The movie version
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.]
Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, A Meta-Algorithm for Creating Fast Algorithms for Counting ON Cells in Odd-Rule Cellular Automata, arXiv:1503.01796 [math.CO], 2015; see also the Accompanying Maple Package.
Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, Odd-Rule Cellular Automata on the Square Grid, arXiv:1503.04249 [math.CO], 2015.
Po-Yi Huang and Wen-Fong Ke, Sequences Derived from The Symmetric Powers of {1, 2, ..., k}, arXiv:2307.07733 [math.CO], 2023.
Tanya Khovanova, There are no coincidences, arXiv preprint 1410.2193 [math.CO], 2014.
Tanya Khovanova and Joshua Xiong, Nim Fractals, arXiv:1405.594291 [math.CO] (2014), p. 10. and J. Int. Seq. 17 (2014) # 14.7.8.
T. Pisanski and T. W. Tucker, Growth in Repeated Truncations of Maps, Atti. Sem. Mat. Fis. Univ. Modena, Vol. 49 (2001), 167-176. (preprint)
Omar E. Pol, Illustration of initial terms: Fig. 1. Neighbors of the vertices, Fig. 2. Overlapping squares, Fig. 3. One-step bishop, (Nov 06 2009).
N. J. A. Sloane, Illustration of a(15) = 81 corresponding to number of ON cells in Odd-rule 013 CA at generation 15
N. J. A. Sloane, On the No. of ON Cells in Cellular Automata, Video of talk in Doron Zeilberger's Experimental Math Seminar at Rutgers University, Feb. 05 2015: Part 1, Part 2
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015.
Ralf Stephan, Divide-and-conquer generating functions. I. Elementary sequences, arXiv:math/0307027 [math.CO], 2003.
Index entries for sequences related to cellular automata
Index entries for sequences that are fixed points of mappings

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.
A generalization of A001316. Cf. A102376.
Partial sums give A130665. - David Applegate, Jun 11 2009
Cf. A000079, A122018, A166453, A268240.

a048883 = a000244 . a000120  -- Reinhard Zumkeller, Nov 14 2011

Nest[ Join[#, 3#] &, {1}, 6] (* Robert G. Wilson v, Jan 24 2006 and modified Jul 27 2014*)
a[n_] := 3^DigitCount[n, 2, 1]; Array[a, 80, 0] (* Jean-François Alcover, Nov 15 2017 *)

PARI
```
a(n)=n=binary(n);3^sum(i=1,#n,n[i])
```

a(n) = Product_{k=0..log_2(n)} 3^b(n,k), where b(n,k) = coefficient of 2^k in binary expansion of n (offset 0). - Paul D. Hanna
a(n) = 3*a(n/2) if n is even, otherwise a(n) = a((n+1)/2).
G.f.: Product_{k>=0} (1+3*x^(2^k)). The generalization k^A000120 has generating function (1 + kx)*(1 + kx^2)*(1 + kx^4)*...
a(n+1) = Sum_{i=0..n} (binomial(n, i) mod 2) * Sum_{j=0..i} (binomial(i, j) mod 2). - Benoit Cloitre, Nov 16 2003
a(0)=1, a(n) = 3*a(n-A053644(n)) for n > 0. - Joe Slater, Jan 31 2016
G.f. A(x) satisfies: A(x) = (1 + 3*x) * A(x^2). - Ilya Gutkovskiy, Jul 09 2019

Corrected by Ralf Stephan, Jun 19 2003
Entry revised by N. J. A. Sloane, May 30 2009
Offset changed to 0, Jun 11 2009

A081603 Number of 2's in ternary representation of n.

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 1, 1, 2, 0, 0, 1, 0, 0, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 2, 2, 3, 0, 0, 1, 0, 0, 1, 1, 1, 2, 0, 0, 1, 0, 0, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 2, 2, 3, 1, 1, 2, 1, 1, 2, 2, 2, 3, 1, 1, 2, 1, 1, 2, 2, 2, 3, 2, 2, 3, 2, 2, 3, 3, 3, 4, 0, 0, 1, 0, 0, 1, 1, 1, 2, 0, 0, 1, 0, 0, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2

Offset: 0

Reinhard Zumkeller, Mar 23 2003

Fixed point of the morphism: 0 ->001; 1 ->112; 2 ->223; 3 ->334, etc., starting from a(0)=0. - Philippe Deléham, Oct 26 2011

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
F. T. Adams-Watters and F. Ruskey, Generating Functions for the Digital Sum and Other Digit Counting Sequences, JIS 12 (2009) 09.5.6.
Eric Weisstein's World of Mathematics, Ternary.

Cf. A007089, A074940, A005836, A081610, A081611, A117592 (2^a(n)).

a081603 0 = 0
a081603 n = a081603 n' + m `div` 2 where (n',m) = divMod n 3
-- Reinhard Zumkeller, Feb 21 2013

A081603 := proc(n)
    local a,d ;
    a := 0 ;
    for d in convert(n,base,3) do
        if d= 2 then
            a := a+1 ;
        end if;
    end do:
    a;
end proc: # R. J. Mathar, Jul 12 2016

Table[Count[IntegerDigits[n,3],2],{n,0,6!}] (* Vladimir Joseph Stephan Orlovsky, Jul 25 2009 *)
Nest[ Flatten[# /. a_Integer -> {a, a, a + 1}] &, {0}, 5] (* Robert G. Wilson v, May 20 2014 *)
DigitCount[Range[0,120],3,2] (* Harvey P. Dale, Jul 10 2019 *)

a(n)=hammingweight(digits(n,3)\2); \\ Ruud H.G. van Tol, Dec 10 2023

from gmpy2 import digits
def A081603(n): return digits(n,3).count('2') # Chai Wah Wu, Dec 05 2024

a(n) = floor(n/2) if n < 3, otherwise a(floor(n/3)) + floor((n mod 3)/2).
A077267(n) + A062756(n) + a(n) = A081604(n);
a(n) = (A053735(n) - A062756(n))/2.

A020654 Lexicographically earliest infinite increasing sequence of nonnegative numbers containing no 5-term arithmetic progression.

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 7, 8, 10, 11, 12, 13, 15, 16, 17, 18, 25, 26, 27, 28, 30, 31, 32, 33, 35, 36, 37, 38, 40, 41, 42, 43, 50, 51, 52, 53, 55, 56, 57, 58, 60, 61, 62, 63, 65, 66, 67, 68, 75, 76, 77, 78, 80, 81, 82, 83, 85, 86, 87, 88, 90, 91, 92, 93, 125, 126, 127

Offset: 1

David W. Wilson

This is also the set of numbers with no "4" in their base-5 representation. In fact, for any prime p, the sequence consisting of numbers with no (p-1) in their base-p expansion is the same as the earliest sequence containing no p-term arithmetic progression. - Nathaniel Johnston, Jun 26-27 2011

Nathaniel Johnston, Table of n, a(n) for n = 1..10000
Robert Baillie and Thomas Schmelzer, Summing Kempner's Curious (Slowly-Convergent) Series, Mathematica Notebook kempnerSums.nb, Wolfram Library Archive, 2008.
J. L. Gerver and L. T. Ramsey, Sets of integers with no long arithmetic progressions generated by the greedy algorithm, Math. Comp., 33 (1979), 1353-1359.
Samuel S. Wagstaff, Jr., On k-free sequences of integers, Math. Comp., 26 (1972), 767-771.
Index entries for 5-automatic sequences.

Cf. A023717.
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).

function a(n)
    m, r, b = n, 0, 1
    while m > 0
        m, q = divrem(m, 4)
        r += b * q
        b *= 5
    end
r end; [a(n) for n in 0:66] |> println # Peter Luschny, Jan 03 2021

seq(`if`(numboccur(4,convert(n,base,5))=0,n,NULL),n=0..127); # Nathaniel Johnston, Jun 27 2011

Select[ Range[ 0, 100 ], (Count[ IntegerDigits[ #, 5 ], 4 ]==0)& ]
Select[Range[0, 120], DigitCount[#, 5, 4] == 0 &] (* Amiram Eldar, Apr 14 2025 *)

is(n)=while(n>4, if(n%5==4, return(0)); n\=5); 1 \\ Charles R Greathouse IV, Feb 12 2017

from sympy.ntheory.factor_ import digits
print([n for n in range(201) if digits(n, 5)[1:].count(4)==0]) # Indranil Ghosh, May 23 2017

from gmpy2 import digits
def A020654(n): return int(digits(n-1,4),5) # Chai Wah Wu, May 06 2025

Sum_{n>=2} 1/a(n) = 7.7794910022243020875287956248411192066951785182667316905881486574421016471305408306837031955619272391023... (calculated using Baillie and Schmelzer's kempnerSums.nb, see Links). - Amiram Eldar, Apr 14 2025

Added "infinite" to definition. - N. J. A. Sloane, Sep 28 2019

A117966 Balanced ternary enumeration (based on balanced ternary representation) of integers; write n in ternary and then replace 2's with (-1)'s.

Original entry on oeis.org

0, 1, -1, 3, 4, 2, -3, -2, -4, 9, 10, 8, 12, 13, 11, 6, 7, 5, -9, -8, -10, -6, -5, -7, -12, -11, -13, 27, 28, 26, 30, 31, 29, 24, 25, 23, 36, 37, 35, 39, 40, 38, 33, 34, 32, 18, 19, 17, 21, 22, 20, 15, 16, 14, -27, -26, -28, -24, -23, -25, -30, -29, -31, -18, -17, -19, -15, -14, -16, -21, -20, -22, -36

Offset: 0

Franklin T. Adams-Watters, Apr 05 2006

As the graph demonstrates, there are large discontinuities in the sequence between terms 3^i-1 and 3^i, and between terms 2*3^i-1 and 2*3^i. - N. J. A. Sloane, Jul 03 2016

			7 in base 3 is 21; changing the 2 to a (-1) gives (-1)*3+1 = -2, so a(7) = -2. I.e., the number of -2 according to the balanced ternary enumeration is 7, which can be obtained by replacing every -1 by 2 in the balanced ternary representation (or expansion) of -2, which is -1,1.

D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 2, pp. 173-175; 2nd. ed. pp. 190-193.

Gheorghe Coserea, Table of n, a(n) for n = 0..59048 (first 729 terms from A. Karttunen)
Ken Levasseur, The Balanced Ternary Number System
Tilman Piesk, First 27 numbers with their ternary representation
N. J. A. Sloane and Brady Haran, Amazing Graphs II (including Star Wars), Numberphile video (2019)
Wikipedia, Balanced ternary

Cf. A117967, A117968, A001057, A004488, A134028, A274107, A059095, A005836, A289814, A244042.

Column k=1 of A319047.

f:= proc(n) local L,i;
   L:= subs(2=-1,convert(n,base,3));
   add(L[i]*3^(i-1),i=1..nops(L))
end proc:
map(f, [$0..100]);
# alternate:
N:= 100: # to get a(0) to a(N)
g:= 0:
for n from 1 to ceil(log[3](N+1)) do
g:= convert(series(3*subs(x=x^3,g)*(1+x+x^2)+x/(1+x+x^2),x,3^n+1),polynom);
od:
seq(coeff(g,x,j),j=0..N); # Robert Israel, Nov 17 2015
# third Maple program:
a:= proc(n) option remember; `if`(n=0, 0,
      3*a(iquo(n, 3, 'r'))+`if`(r=2, -1, r))
    end:
seq(a(n), n=0..3^4-1);  # Alois P. Heinz, Aug 14 2019

Map[FromDigits[#, 3] &, IntegerDigits[#, 3] /. 2 -> -1 & /@ Range@ 80] (* Michael De Vlieger, Nov 17 2015 *)

a(n) = subst(Pol(apply(x->if(x == 2, -1, x), digits(n,3)), 'x), 'x, 3)
vector(73, i, a(i-1))  \\ Gheorghe Coserea, Nov 17 2015

def a(n):
    if n==0: return 0
    if n%3==0: return 3*a(n//3)
    elif n%3==1: return 3*a((n - 1)//3) + 1
    else: return 3*a((n - 2)//3) - 1
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 06 2017

a(0) = 0, a(3n) = 3a(n), a(3n+1) = 3a(n)+1, a(3n+2) = 3a(n)-1.
G.f. satisfies A(x) = 3*A(x^3)*(1+x+x^2) + x/(1+x+x^2). - corrected by Robert Israel, Nov 17 2015
A004488(n) = a(n)^{-1}(-a(n)). I.e., if a(n) <= 0, A004488(n) = A117967(-a(n)) and if a(n) > 0, A004488(n) = A117968(a(n)).
a(n) = n - 3 * A005836(A289814(n) + 1). - Andrey Zabolotskiy, Nov 11 2019

Name corrected by Andrey Zabolotskiy, Nov 10 2019

A289813 A binary encoding of the ones in ternary representation of n (see Comments for precise definition).

Original entry on oeis.org

0, 1, 0, 2, 3, 2, 0, 1, 0, 4, 5, 4, 6, 7, 6, 4, 5, 4, 0, 1, 0, 2, 3, 2, 0, 1, 0, 8, 9, 8, 10, 11, 10, 8, 9, 8, 12, 13, 12, 14, 15, 14, 12, 13, 12, 8, 9, 8, 10, 11, 10, 8, 9, 8, 0, 1, 0, 2, 3, 2, 0, 1, 0, 4, 5, 4, 6, 7, 6, 4, 5, 4, 0, 1, 0, 2, 3, 2, 0, 1, 0, 16

Offset: 0

Rémy Sigrist, Jul 12 2017

The ones in the binary representation of a(n) correspond to the ones in the ternary representation of n; for example: ternary(42) = 1120 and binary(a(42)) = 1100 (a(42) = 12).
See A289814 for the sequence encoding the twos in ternary representation of n.
By design, a(n) AND A289814(n) = 0 (where AND stands for the bitwise AND operator).
See A289831 for the sum of this sequence and A289814.
For each pair of numbers without common bits in base 2 representation, say x and y, there is a unique index, say n, such that a(n) = x and A289814(n) = y; in fact, n = A289869(x,y).
The scatterplot of this sequence vs A289814 looks like a Sierpinski triangle pivoted to the side.
For any t > 0: we can adapt the algorithm used here and in A289814 in order to uniquely enumerate every tuple of t numbers mutually without common bits in base 2 representation.

			The first values, alongside the ternary representation of n, and the binary representation of a(n), are:
n       a(n)    ternary(n)  binary(a(n))
--      ----    ----------  ------------
0       0       0           0
1       1       1           1
2       0       2           0
3       2       10          10
4       3       11          11
5       2       12          10
6       0       20          0
7       1       21          1
8       0       22          0
9       4       100         100
10      5       101         101
11      4       102         100
12      6       110         110
13      7       111         111
14      6       112         110
15      4       120         100
16      5       121         101
17      4       122         100
18      0       200         0
19      1       201         1
20      0       202         0
21      2       210         10
22      3       211         11
23      2       212         10
24      0       220         0
25      1       221         1
26      0       222         0

Rémy Sigrist, Table of n, a(n) for n = 0..6560

Cf. A000120, A004488, A005836, A053735, A289814, A289831, A289869.

Table[FromDigits[#, 2] &[IntegerDigits[n, 3] /. 2 -> 0], {n, 0, 81}] (* Michael De Vlieger, Jul 20 2017 *)

a(n) = my (d=digits(n,3)); fromdigits(vector(#d, i, if (d[i]==1, 1, 0)), 2)

a(n) = fromdigits(digits(n, 3)%2, 2); \\ Ruud H.G. van Tol, May 08 2024

from sympy.ntheory.factor_ import digits
def a(n):
    d = digits(n, 3)[1:]
    return int("".join('1' if i==1 else '0' for i in d), 2)
print([a(n) for n in range(51)]) # Indranil Ghosh, Jul 20 2017

a(0) = 0.
a(3*n) = 2 * a(n).
a(3*n+1) = 2 * a(n) + 1.
a(3*n+2) = 2 * a(n).
Also, a(n) = A289814(A004488(n)).
A053735(n) = A000120(a(n)) + 2*A000120(A289814(n)). - Antti Karttunen, Jul 20 2017

A102376 a(n) = 4^A000120(n).

Original entry on oeis.org

1, 4, 4, 16, 4, 16, 16, 64, 4, 16, 16, 64, 16, 64, 64, 256, 4, 16, 16, 64, 16, 64, 64, 256, 16, 64, 64, 256, 64, 256, 256, 1024, 4, 16, 16, 64, 16, 64, 64, 256, 16, 64, 64, 256, 64, 256, 256, 1024, 16, 64, 64, 256, 64, 256, 256, 1024, 64, 256, 256, 1024, 256, 1024, 1024

Offset: 0

Paul Barry, Jan 05 2005

Consider a simple cellular automaton, a grid of binary cells c(i,j), where the next state of the grid is calculated by applying the following rule to each cell: c(i,j) = ( c(i+1,j-1) + c(i+1,j+1) + c(i-1,j-1) + c(i-1,j+1) ) mod 2 If we start with a single cell having the value 1 and all the others 0, then the aggregate values of the subsequent states of the grid will be the terms in this sequence. - Andras Erszegi (erszegi.andras(AT)chello.hu), Mar 31 2006. See link for initial states. - N. J. A. Sloane, Feb 12 2015
This is the odd-rule cellular automaton defined by OddRule 033 (see Ekhad-Sloane-Zeilberger "Odd-Rule Cellular Automata on the Square Grid" link). - N. J. A. Sloane, Feb 25 2015
First differences of A116520. - Omar E. Pol, May 05 2010

			1 + 4*x + 4*x^2 + 16*x^3 + 4*x^4 + 16*x^5 + 16*x^6 + 64*x^7 + 4*x^8 + ...
From _Omar E. Pol_, Jun 07 2009: (Start)
Triangle begins:
  1;
  4;
  4,16;
  4,16,16,64;
  4,16,16,64,16,64,64,256;
  4,16,16,64,16,64,64,256,16,64,64,256,64,256,256,1024;
  4,16,16,64,16,64,64,256,16,64,64,256,64,256,256,1024,16,64,64,256,64,256,...
(End)

N. J. A. Sloane, Table of n, a(n) for n = 0..10000
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.]
Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, A Meta-Algorithm for Creating Fast Algorithms for Counting ON Cells in Odd-Rule Cellular Automata, arXiv:1503.01796 [math.CO], 2015; see also the Accompanying Maple Package.
Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, Odd-Rule Cellular Automata on the Square Grid, arXiv:1503.04249 [math.CO], 2015.
Nathan Epstein, Animation of CA generating A102376
N. J. A. Sloane, On the No. of ON Cells in Cellular Automata, Video of talk in Doron Zeilberger's Experimental Math Seminar at Rutgers University, Feb. 05 2015: Part 1, Part 2
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015.
N. J. A. Sloane, Illustration of generations 0-15 of the cellular automaton
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Alexander Yu. Vlasov, Modelling reliability of reversible circuits with 2D second-order cellular automata, arXiv:2312.13034 [nlin.CG], 2023. See page 13.
Index entries for sequences related to cellular automata

For generating functions Prod_{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.
A151783 is a very similar sequence.
Cf. A001316, A048883, A000079, A116520, A000302.
See A160239 for the analogous CA defined by Rule 204 on an 8-celled neighborhood.

a102376 = (4 ^) . a000120  -- Reinhard Zumkeller, Feb 13 2015

seq(4^convert(convert(n,base,2),`+`),n=0..100); # Robert Israel, Apr 30 2017

Table[4^DigitCount[n, 2, 1], {n, 0, 100}] (* Indranil Ghosh, Apr 30 2017 *)

{a(n) = if( n<0, 0, 4^subst( Pol( binary(n)), x, 1))} /* Michael Somos, May 29 2008 */
a(n) = 4^hammingweight(n); \\ Michel Marcus, Apr 30 2017

def a(n): return 4**bin(n)[2:].count("1") # Indranil Ghosh, Apr 30 2017

def A102376(n): return 1<<(n.bit_count()<<1) # Chai Wah Wu, Nov 15 2022

Formulas due to Paul D. Hanna: (Start)
G.f.: Product_{k>=0} 1 + 4x^(2^k).
a(n) = Product_{k=0..log_2(n)} 4^b(n, k), b(n, k)=coefficient of 2^k in binary expansion of n.
a(n) = Sum_{k=0..n} (C(n, k) mod 2)*3^A000120(n-k). (End)
a(n) = Sum_{k=0..n} (C(n, k) mod 2) * Sum_{j=0..k} (C(k, j) mod 2) * Sum_{i=0..j} (C(j, i) mod 2). - Paul Barry, Apr 01 2005
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = w * (u^2 - 2*u*v + 5*v^2) - 4*v^3. - Michael Somos, May 29 2008
Run length transform of A000302. - N. J. A. Sloane, Feb 23 2015

A020657 Lexicographically earliest increasing sequence of nonnegative numbers that contains no arithmetic progression of length 7.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25, 26, 28, 29, 30, 31, 32, 33, 35, 36, 37, 38, 39, 40, 49, 50, 51, 52, 53, 54, 56, 57, 58, 59, 60, 61, 63, 64, 65, 66, 67, 68, 70, 71, 72, 73, 74, 75, 77, 78, 79, 80, 81, 82, 84, 85

Offset: 1

David W. Wilson

Also the set of numbers with no "6" in their base-7 representation; see Gerver-Ramsey, also comments in A020654. - Nathaniel Johnston, Jun 27 2011

Up to the offset, identical to A037470. There are lexicographically earlier, but non-monotonic sequences which do not contain a 7-term AP, e.g., starting with 0,0,0,0,0,0,1,0,... - M. F. Hasler, Oct 05 2014

Nathaniel Johnston, Table of n, a(n) for n = 1..10000
J. L. Gerver and L. T. Ramsey, Sets of integers with no long arithmetic progressions generated by the greedy algorithm, Math. Comp., 33 (1979), 1353-1359.

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).

seq(`if`(numboccur(6,convert(n,base,7))=0,n,NULL),n=0..85); # Nathaniel Johnston, Jun 27 2011

Select[Range[0, 100], FreeQ[IntegerDigits[#, 7], 6]&] (* Jean-François Alcover, Jan 27 2023 *)

a(n)=vector(#n=digits(n-1, 6), i, 7^(#n-i))*n~ \\ M. F. Hasler, Oct 05 2014

from gmpy2 import digits
def A020657(n): return int(digits(n-1,6),7) # Chai Wah Wu, May 06 2025

Name edited by M. F. Hasler, Oct 10 2014. Further edited by N. J. A. Sloane, Jan 04 2016

cp's OEIS Frontend

A235266 Primes whose base-2 representation is also the base-3 representation of a prime.

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A005823 Numbers whose ternary expansion contains no 1's.

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A011539 "9ish numbers": decimal representation contains at least one nine.

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A048883 a(n) = 3^wt(n), where wt(n) = A000120(n).

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A081603 Number of 2's in ternary representation of n.

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A020654 Lexicographically earliest infinite increasing sequence of nonnegative numbers containing no 5-term arithmetic progression.

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