cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

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

Views

Author

N. J. A. Sloane, Jeffrey Shallit

Keywords

Comments

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

Examples

			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

References

  • 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).

Links

Crossrefs

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.

Programs

  • Haskell
    a005836 n = a005836_list !! (n-1)
a005836_list = filter ((== 1) . a039966) [0..]
-- Reinhard Zumkeller, Jun 09 2012, Sep 29 2011
  • Julia
    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
  • Maple
    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
  • Mathematica
    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 *)
  • PARI
    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
  • PARI
    is(n)=while(n,if(n%3>1,return(0));n\=3);1 \\ Charles R Greathouse IV, Mar 07 2013
  • PARI
    a(n) = fromdigits(binary(n-1),3);  \\ Gheorghe Coserea, Jun 15 2018
  • Python
    def A005836(n):
    return int(format(n-1,'b'),3) # Chai Wah Wu, Jan 04 2015

Formula

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

Extensions

Offset corrected by N. J. A. Sloane, Mar 02 2008
Edited by the Associate Editors of the OEIS, Apr 07 2009

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

Views

Author

N. J. A. Sloane, Jeffrey Shallit

Keywords

Comments

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

References

  • 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).

Links

Crossrefs

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

Programs

  • Maple
    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
  • Mathematica
    Select[ Range[ 0, 729 ], (Count[ IntegerDigits[ #, 3 ], 1 ]==0)& ]
Select[Range[0,700],DigitCount[#,3,1]==0&] (* Harvey P. Dale, Mar 12 2016 *)
  • PARI
    is(n)=while(n,if(n%3==1,return(0),n\=3));1 \\ Charles R Greathouse IV, Apr 20 2012
  • PARI
    a(n)=n=binary(n-1);sum(i=1,#n,2*n[i]*3^(#n-i)) \\ Charles R Greathouse IV, Apr 20 2012
  • PARI
    a(n)=2*fromdigits(binary(n-1),3) \\ Charles R Greathouse IV, Aug 24 2016
  • Python
    def A005823(n):
    return 2*int(format(n-1,'b'),3) # Chai Wah Wu, Jan 04 2015

Formula

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

Extensions

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.

A125289 Numbers with unique nonzero digit in decimal representation.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 20, 22, 30, 33, 40, 44, 50, 55, 60, 66, 70, 77, 80, 88, 90, 99, 100, 101, 110, 111, 200, 202, 220, 222, 300, 303, 330, 333, 400, 404, 440, 444, 500, 505, 550, 555, 600, 606, 660, 666, 700, 707, 770, 777, 800, 808, 880, 888, 900, 909
Offset: 1

Views

Author

Reinhard Zumkeller, Nov 26 2006

Keywords

Comments

A043537(a(n)) <= 2.
A043537(A004719(a(n))) = 1: A004719(a(n)) is a repdigit number, see A010785;
also numbers having exactly one partition into digit values of their decimal representations: A061827(a(n))=1.

Links

Crossrefs

Cf. A125292.

Programs

  • PARI
    is(n, base=10) = #Set(select(sign, digits(n, base)))==1 \\ Rémy Sigrist, Mar 28 2020
  • PARI
    a(n,base=10) = { for (w=0, oo, if (n<=(base-1)*2^w, my (d=1+(n-1)\2^w, k=2^w+(n-1)%(2^w)); return (d*fromdigits(binary(k), base)), n -= (base-1)*2^w)) } \\ Rémy Sigrist, Mar 28 2020
  • Python
    A125289_list = [n for n in range(10**4) if len(set(str(n))-{'0'})==1]
# Chai Wah Wu, Jan 04 2015
  • Python
    from itertools import count, product, islice
def A125289_gen(): # generator of terms
    yield from (int(d+''.join(m)) for l in count(0) for d in '123456789' for m in product('0'+d,repeat=l))
A125289_list = list(islice(A125289_gen(),20)) # Chai Wah Wu, Mar 14 2025

A125291 Number of partitions of n into positive digit values of its ternary representation.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 4, 1, 1, 1, 6, 1, 1, 8, 8, 9, 9, 1, 10, 1, 11, 12, 12, 1, 13, 1, 1, 1, 15, 1, 1, 17, 17, 18, 18, 1, 1, 20, 1, 1, 21, 22, 22, 23, 23, 24, 24, 25, 25, 26, 26, 27, 27, 1, 28, 1, 29, 30, 30, 1, 31, 1, 32, 33, 33, 34, 34, 35, 35, 36, 36, 1, 37, 1, 38, 39, 39, 1, 40, 1, 1, 1, 42
Offset: 1

Views

Author

Reinhard Zumkeller, Nov 26 2006

Keywords

Comments

a(A125292(n))=1; a(A125293(n))=floor((n+2)/2)=A008619(n).

Links

Crossrefs

Cf. A007089, A061827.

Formula

a(n) = 1 + floor(n/2) * (1 - 0^(A062756(n)*A081603(n))).

A125293 Numbers with at least one 1 and one 2 in ternary representation.

Original entry on oeis.org

5, 7, 11, 14, 15, 16, 17, 19, 21, 22, 23, 25, 29, 32, 33, 34, 35, 38, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 55, 57, 58, 59, 61, 63, 64, 65, 66, 67, 68, 69, 70, 71, 73, 75, 76, 77, 79, 83, 86, 87, 88, 89, 92, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106
Offset: 1

Views

Author

Reinhard Zumkeller, Nov 26 2006

Keywords

Comments

Complement of A125292; A062756(a(n))*A081603(a(n)) > 0;
A125291(a(n)) > 1.

Links

Crossrefs

Cf. A007089, A125290.

Programs

  • Mathematica
    Select[Range[120],DigitCount[#,3,1]>0&&DigitCount[#,3,2]>0&] (* Harvey P. Dale, Apr 12 2013 *)

A372317 For any n > 0 with leading ternary digit h = A122586(n), reverse digits in blocks in ternary expansion of n where blocks are separated by h's; a(0) = 0.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 15, 12, 13, 14, 11, 16, 17, 18, 21, 20, 19, 22, 23, 24, 25, 26, 27, 28, 45, 30, 31, 32, 33, 46, 51, 36, 37, 42, 39, 40, 41, 38, 43, 44, 29, 34, 47, 48, 49, 50, 35, 52, 53, 54, 63, 56, 57, 66, 65, 60, 61, 62, 55, 64, 59, 58, 67
Offset: 0

Views

Author

Rémy Sigrist, Apr 27 2024

Keywords

Comments

This sequence is a self-inverse permutation of the nonnegative integers.
This sequence has infinitely many fixed points (A125292, among others).

Examples

			For n = 5323: the ternary expansion of 5323 is "21022011", the corresponding leading digit is "2", we have three blocks: "10", "" and "011", their reversals are: "01", "" and "110", so the ternary expansion of a(5323) is "20122110", and a(5323) = 4845.

Links

Crossrefs

See A367307 and A372318 for similar sequences.
Cf. A122586, A125292.

Programs

  • PARI
    a(n, base = 3) = { my (d = digits(n, base), i = 1); for (j = 2, #d+1, if (j==#d+1 || d[i]==d[j], my (ii = i+1, jj = j-1); while (ii < jj, [d[ii], d[jj]] = [d[jj], d[ii]]; ii++; jj--;); i = j;);); fromdigits(d, base); }
Showing 1-6 of 6 results.

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