A007089 -id:A007089 - cp's OEIS frontend

A033016 Numbers whose base-3 expansion has no run of digits with length < 2.

4, 8, 13, 26, 36, 40, 44, 72, 76, 80, 108, 117, 121, 125, 134, 216, 229, 234, 238, 242, 324, 328, 332, 351, 360, 364, 368, 377, 396, 400, 404, 648, 652, 656, 684, 688, 692, 702, 715, 720, 724, 728, 972, 976, 980, 985, 998, 1053

Offset: 1

Clark Kimberling

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A007089. Supersequence of A033001.

Select[Range[10000], Min[Length/@Split[IntegerDigits[#, 3]]]>1&] (* Vincenzo Librandi, Feb 05 2014 *)

A037314 Numbers whose base-3 and base-9 expansions have the same digit sum.

Original entry on oeis.org

0, 1, 2, 9, 10, 11, 18, 19, 20, 81, 82, 83, 90, 91, 92, 99, 100, 101, 162, 163, 164, 171, 172, 173, 180, 181, 182, 729, 730, 731, 738, 739, 740, 747, 748, 749, 810, 811, 812, 819, 820, 821, 828, 829, 830, 891, 892, 893, 900, 901, 902, 909, 910, 911

Offset: 0

Clark Kimberling

a(n) = Sum_{i=0..m} d(i)*9^i, where Sum_{i=0..m} d(i)*3^i is the base-3 representation of n.
Numbers that can be written using only digits 0, 1 and 2 in base 9. Also, write n in base 3, read as base 9: (3) [n] (9) in base change notation. a(3n+k) = 9a(n)+k for k in {0,1,2}. - Franklin T. Adams-Watters, Jul 24 2006
Also, every term k corresponds to a unique pair i,j with k = a(i) + 3*a(j) (similarly to the Moser-de Bruijn sequence). - Luis Rato, May 02 2024

Cf. A007089, A208665, A338086 (ternary digit duplication).

Cf. A053735, A053830.

function a(n)
    m, r, b = n, 0, 1
    while m > 0
        m, q = divrem(m, 3)
        r += b * q
        b *= 9
    end
r end
[a(n) for n in 0:53] |> println # Peter Luschny, Jan 03 2021

Table[FromDigits[RealDigits[n, 3], 9], {n, 1, 100}] (* Clark Kimberling, Aug 14 2012 *)
Select[Range[0,1000],Total[IntegerDigits[#,3]]==Total[IntegerDigits[#,9]]&] (* Harvey P. Dale, Feb 17 2020 *)

a(n) = {my(d = digits(n, 3)); subst(Pol(d), x, 9);} \\ Michel Marcus, Apr 09 2015

G.f. f(x) = Sum_{j>=0} 9^j*x^(3^j)*(1+x^(3^j)-2*x^(2*3^j))/((1-x)*(1-x^(3^(j+1)))) satisfies f(x) = 9*(x^2+x+1)*f(x^3) + x*(1+2*x)/(1-x^3). - Robert Israel, Apr 13 2015

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Jun 08 2007

Offset changed to 0 by Clark Kimberling, Aug 14 2012

A081605 Numbers having at least one 0 in their ternary representation.

Original entry on oeis.org

0, 3, 6, 9, 10, 11, 12, 15, 18, 19, 20, 21, 24, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 42, 45, 46, 47, 48, 51, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 69, 72, 73, 74, 75, 78, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100

Offset: 1

Reinhard Zumkeller, Mar 23 2003

Complement of A032924.

A212193(a(n)) <> 0. [Reinhard Zumkeller, May 04 2012]

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A007089, A077267, A081607, A081606, A074940.

import Data.List (findIndices)
a081605 n = a081605_list !! (n-1)
a081605_list = findIndices (/= 0) a212193_list
-- Reinhard Zumkeller, May 04 2012

Select[Range[0,100],DigitCount[#,3,0]>0&] (* Harvey P. Dale, Aug 10 2021 *)

from itertools import count, islice
def A081605_gen(): # generator of terms
    a = -1
    for n in count(2):
        b = int(bin(n)[3:],3) + (3**(n.bit_length()-1)-1>>1)
        yield from range(a+1,b)
        a = b
A081605_list = list(islice(A081605_gen(),30)) # Chai Wah Wu, Oct 13 2023

A118594 Palindromes in base 3 (written in base 3).

Original entry on oeis.org

0, 1, 2, 11, 22, 101, 111, 121, 202, 212, 222, 1001, 1111, 1221, 2002, 2112, 2222, 10001, 10101, 10201, 11011, 11111, 11211, 12021, 12121, 12221, 20002, 20102, 20202, 21012, 21112, 21212, 22022, 22122, 22222, 100001, 101101, 102201, 110011, 111111, 112211, 120021

Offset: 1

Martin Renner, May 08 2006

The number of n-digit terms is given by A225367. - M. F. Hasler, May 05 2013 [Moved here on May 08 2013]
Digit-wise application of A000578 (and also superposition of a(n) with its horizontal OR vertical reflection) yields A006072. - M. F. Hasler, May 08 2013
Equivalently, palindromes k (written in base 10) such that 4*k is a palindrome. - Bruno Berselli, Sep 12 2018

Michael S. Branicky, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Palindromic Number
Eric Weisstein's World of Mathematics, Ternary

Cf. A007089, A014190, A057148, A118595, A118596, A118597, A118598, A118599, A118600, A002113.

(* get NextPalindrome from A029965 *) Select[NestList[NextPalindrome, 0, 1110], Max@IntegerDigits@# < 3 &] (* Robert G. Wilson v, May 09 2006 *)
Select[FromDigits/@Tuples[{0,1,2},8],IntegerDigits[#]==Reverse[ IntegerDigits[ #]]&] (* Harvey P. Dale, Apr 20 2015 *)

{for(l=1,5,u=vector((l+1)\2,i,10^(i-1)+(2*i-11&&i==1,2]), print1(v*u",")))} \\ The n-th term could be produced by using (partial sums of) A225367 to skip all shorter terms, and then skipping the adequate number of vectors v until n is reached.  - M. F. Hasler, May 08 2013

from itertools import count, islice, product
def agen(): # generator of terms
    yield from [0, 1, 2]
    for d in count(2):
        for start in "12":
            for rest in product("012", repeat=d//2-1):
                left = start + "".join(rest)
                for mid in [[""], ["0", "1", "2"]][d%2]:
                    yield int(left + mid + left[::-1])
print(list(islice(agen(), 42))) # Michael S. Branicky, Mar 29 2022

from sympy import integer_log
from gmpy2 import digits
def A118594(n):
    if n == 1: return 0
    y = 3*(x:=3**integer_log(n>>1,3)[0])
    return int((s:=digits(n-x,3))+s[-2::-1] if nChai Wah Wu, Jun 14 2024

[int(n.str(base=3)) for n in (0..757) if Word(n.digits(3)).is_palindrome()] # Peter Luschny, Sep 13 2018

More terms from Robert G. Wilson v, May 09 2006

a(40) and beyond from Michael S. Branicky, Mar 29 2022

A136808 Numbers k such that k and k^2 use only the digits 0, 1 and 2.

Original entry on oeis.org

0, 1, 10, 11, 100, 101, 110, 1000, 1001, 1010, 1011, 1100, 1101, 10000, 10001, 10010, 10011, 10100, 10110, 11000, 11001, 11010, 100000, 100001, 100010, 100011, 100100, 100101, 100110, 101000, 101001, 101100, 110000, 110001, 110010, 110100, 1000000, 1000001, 1000010, 1000011, 1000100

Offset: 1

Jonathan Wellons (wellons(AT)gmail.com), Jan 22 2008

Generated with DrScheme.
Subsequence of A136809, A136816, ..., A136836. - M. F. Hasler, Jan 24 2008
A278038(18) = 10101, A136827(294) = 10110001101, A136831(1276) = 101100010001101 resp. A136836(1262) = 101090009991101 are the first terms from where on these four sequences differ from the present one. - M. F. Hasler, Nov 15 2017
From Jovan Radenkovicc, Nov 15 2024: (Start)
A nonnegative integer n is in this sequence iff 10*n is also in this sequence.
Not a subsequence of A278038 (binary numbers without '111'). A counterexample is 10^2884 + 10^2880 + 10^2872 + 10^2857 + 10^2497 + 10^2426 + 10^2285 + 10^2004 + 10^1443 + 10^1442 + 10^1441 + 10^881 + 10^600 + 10^459 + 10^388 + 10^27 + 10^12 + 10^4 + 1. There are infinitely many counterexamples not divisible by 10. This counterexample follows from the fact that 111^2+2000*4+200*4=12321+8000+800=21121. In fact, every binary substring will eventually occur in this sequence. Also, if n is a term containing only the digits 0 and 1, then 10^k*n+1 and n+10^k are also in this sequence for any sufficiently large integer k. (End)

			101000100100001^2 = 10201020220210222010200200001.

Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 1359 terms from Jonathan Wellons)
Jonathan Wellons, Tables of Shared Digits [archived]
Index to sequences related to squares.

A subsequence of the binary numbers A007088.
Cf. A278038.
Cf. A136809, A136810, ..., A137147 for other digit combinations.
See also A058412 = A058411^2: squares having only digits {0,1,2}, A277946 = A277959^2 = squares whose largest digit is 2.

isA136808 := proc(n) local ndgs,n2dgs ; ndgs := convert(convert(n,base,10),set) ; n2dgs := convert(convert(n^2,base,10),set) ; if ( (ndgs union n2dgs) minus {0,1,2} ) = {} then true ; else false ; fi ; end: LtonRev := proc(L) local i ; add(op(i,L)*10^(i-1),i=1..nops(L)) ; end: A007089 := proc(n) convert(n,base,3) ; LtonRev(%) ; end: n := 1: for i from 0 do n3 := A007089(i) ; if isA136808(n3) then printf("%d %d ",n,n3) ; n := n+1 ; fi ; od: # R. J. Mathar, Jan 24 2008

Select[FromDigits/@Tuples[{0,1},7],Union[Take[DigitCount[#^2],{3,9}]]=={0}&] (* Harvey P. Dale, May 29 2013 *)

for(n=1,999,vecmax(digits((N=fromdigits(binary(n),10))^2))<3 && print1(N",")) \\ M. F. Hasler, Nov 15 2017

A265747 Numbers written in Jacobsthal greedy base.

Original entry on oeis.org

0, 1, 2, 10, 11, 100, 101, 102, 110, 111, 200, 1000, 1001, 1002, 1010, 1011, 1100, 1101, 1102, 1110, 1111, 10000, 10001, 10002, 10010, 10011, 10100, 10101, 10102, 10110, 10111, 10200, 11000, 11001, 11002, 11010, 11011, 11100, 11101, 11102, 11110, 11111, 20000, 100000, 100001, 100002, 100010, 100011, 100100

Offset: 0

Antti Karttunen, Dec 17 2015

These are called "Jacobsthal Representation Numbers" in Horadam's 1996 paper.
Sum_{i=0..} digit(i)*A001045(2+digit(i)) recovers n from such representation a(n), where digit(0) stands for the least significant digit (at the right), and A001045(k) gives the k-th Jacobsthal number.
No larger digits than 2 will occur, which allows representing the same sequence in a more compact form by base-3 coding in A265746.
Sequence A197911 gives the terms with no digit "2" in their representation, while its complement A003158 gives the terms where "2" occurs at least once.
Numbers beginning with digit "2" in this representation are given by A020988(n) [= 2*A002450(n) = 2*A001045(2n)].

			For n=7, when selecting the terms of A001045 with the greedy algorithm, we need terms A001045(4) + A001045(2) + A001045(2) = 5 + 1 + 1, thus a(7) = "102".
For n=10, we need A001045(4) + A001045(4) = 5+5, thus a(10) = "200".

Antti Karttunen, Table of n, a(n) for n = 0..10923
A. F. Horadam, Jacobsthal Representation Numbers, Fib Quart. 34 (1996), 40-54. (See especially page 50, which is page 11 in PDF.)

Cf. A001045, A002450, A020988, A007089, A197911, A003158.
Cf. A265745 (sum of digits).
Cf. A265746 (same numbers interpreted in base-3, then shown in decimal).
Cf. A084639 (positions of repunits).
Cf. A007961, A014417, A014418, A244159 for analogous sequences.

jacob[n_] := (2^n - (-1)^n)/3; maxInd[n_] := Floor[Log2[3*n + 1]]; A265747[n_] := A265747[n] = 10^(maxInd[n] - 2) + A265747[n - jacob[maxInd[n]]]; A265747[0] = 0; Array[A265747, 100, 0] (* Amiram Eldar, Jul 21 2023 *)

A130249(n) = floor(log(3*n + 1) / log(2));
A001045(n) = (2^n - (-1)^n) / 3;
A265747(n) = {if(n==0, 0, my(d=n - A001045(A130249(n))); 10^(A130249(n)-2) + if(d == 0, 0, A265747(d)));} \\ Amiram Eldar, Jul 21 2023

def greedyJ(n): m = (3*n+1).bit_length() - 1; return (m, (2**m-(-1)**m)//3)
def a(n):
    if n == 0: return 0
    place, value = greedyJ(n)
    return 10**(place-2) + a(n - value)
print([a(n) for n in range(49)]) # Michael S. Branicky, Jul 11 2021

a(0) = 0; for n >= 1, a(n) = 10^(A130249(n)-2) + a(n-A001045(A130249(n))).

a(n) = A007089(A265746(n)).

A062891 When expressed in base 3 and then interpreted in base 9, is a multiple of the original number.

Original entry on oeis.org

0, 1, 2, 3, 6, 9, 13, 18, 26, 27, 34, 39, 47, 54, 78, 81, 91, 102, 117, 121, 141, 162, 182, 234, 242, 243, 262, 273, 306, 351, 363, 423, 486, 546, 702, 726, 729, 757, 786, 819, 918, 1048, 1053, 1089, 1093, 1183, 1269, 1458, 1514, 1638, 2106, 2178, 2186, 2187

Offset: 1

Erich Friedman, Jul 21 2001

			13 in base 3 is 111, which interpreted in base 9 is 91 = 7*13.

Harvey P. Dale, Table of n, a(n) for n = 1..751 [offset shifted by _Georg Fischer_, Jun 15 2021]

Cf. A007089 (base 3), A007095 (base 9), A037314 (base 3 -> 9).

Other digit spreads: A062846 (binary), A343550 (decimal).

q:= n-> (l-> n=0 or 0=irem(add(l[i]*9^(i-1),
         i=1..nops(l)), n))(convert(n, base, 3)):
select(q, [$0..3000])[];  # Alois P. Heinz, Apr 20 2021

Join[{0},Select[Range[2200],Divisible[FromDigits[IntegerDigits[#,3],9],#]&]] (* Harvey P. Dale, Apr 11 2017 *)

Offset changed to 1 by Kevin Ryde, Apr 24 2021

A065369 Replace 3^k with (-3)^k in ternary expansion of n.

Original entry on oeis.org

0, 1, 2, -3, -2, -1, -6, -5, -4, 9, 10, 11, 6, 7, 8, 3, 4, 5, 18, 19, 20, 15, 16, 17, 12, 13, 14, -27, -26, -25, -30, -29, -28, -33, -32, -31, -18, -17, -16, -21, -20, -19, -24, -23, -22, -9, -8, -7, -12, -11, -10, -15, -14, -13, -54, -53, -52, -57, -56, -55, -60, -59, -58, -45, -44, -43, -48, -47, -46

Offset: 0

Marc LeBrun, Oct 31 2001

Base 3 representation for n (in lexicographic order) converted from base -3 to base 10.
Notation: (3)[n](-3)
Fixed point of the morphism 0-> 0,1,2 ; 1-> -3,-2,-1 ; 2-> -6,-5,-4 ; ...; n-> -3n,-3n+1,-3n+2. - Philippe Deléham, Oct 22 2011

			15 = +1(9)+2(3)+0(1) -> +1(+9)+2(-3)+0(+1) = +3 = a(15)

Rémy Sigrist, Table of n, a(n) for n = 0..19682
Dana G. Korssjoen, Biyao Li, Stefan Steinerberger, Raghavendra Tripathi, and Ruimin Zhang, Finding structure in sequences of real numbers via graph theory: a problem list, arXiv:2012.04625 [math.CO], 2020-2021.

Cf. A007089, A053985, A065367, A073785, A073791, A073792, A073793, A073794, A073795, A073796, A073835.

f[n_Integer, b_Integer] := Block[{l = IntegerDigits[n]}, Sum[l[[ -i]]*(-b)^(i - 1), {i, 1, Length[l]}]]; a = Table[ FromDigits[ IntegerDigits[n, 3]], {n, 0, 80}]; b = {}; Do[b = Append[b, f[a[[n]], 3]], {n, 1, 80}]; b

a(n) = fromdigits(digits(n, 3), -3) \\ Rémy Sigrist, Feb 06 2020

a(n) = Sum_{k>=0} A030341(n,k)*(-3)^k. - Philippe Deléham, Oct 22 2011

a(3*k+m) = -3*a(k)+m for 0 <= m < 3. - Chai Wah Wu, Jan 16 2020

A255588 Convert n to base 3, move the least significant digit to the most significant one and convert back to base 10.

Original entry on oeis.org

0, 1, 2, 1, 4, 7, 2, 5, 8, 3, 12, 21, 4, 13, 22, 5, 14, 23, 6, 15, 24, 7, 16, 25, 8, 17, 26, 9, 36, 63, 10, 37, 64, 11, 38, 65, 12, 39, 66, 13, 40, 67, 14, 41, 68, 15, 42, 69, 16, 43, 70, 17, 44, 71, 18, 45, 72, 19, 46, 73, 20, 47, 74, 21, 48, 75, 22, 49, 76, 23

Offset: 0

Paolo P. Lava, Feb 27 2015

a(3*n) = n.

Fixed points of the transform are listed in A048328.

			10 in base 3 is 101: moving the least significant digit to the most significant one we have 110 that is 12 in base 10.

Indranil Ghosh, Table of n, a(n) for n = 0..19683 (terms 0..1000 Paolo P. Lava)

Cf. A030102, A038572, A048328, A255589-A255594, A048787, A255689-A255693.

with(numtheory): P:=proc(q,h) local a,b,k,n; print(0);
for n from 1 to q do
a:=convert(n,base,h); b:=[]; for k from 2 to nops(a) do b:=[op(b),a[k]]; od; a:=[op(b),a[1]];
a:=convert(a,base,h,10); b:=0; for k from nops(a) by -1 to 1 do b:=10*b+a[k]; od;
print(b); od; end: P(10^4,3);

roll[n_, b_] := Block[{w = IntegerDigits[n, b]}, Prepend[Most@ w, Last@ w]]; b = 3; FromDigits[#, b] & /@ (roll[#, b] & /@ Range[0, 69]) (* Michael De Vlieger, Mar 04 2015 *)
Table[FromDigits[RotateRight[IntegerDigits[n,3]],3],{n,0,70}] (* Harvey P. Dale, Feb 20 2022 *)

def A255588(n):
    x=A007089(n)
    return int(x[-1]+x[:-1], 3) # Indranil Ghosh, Feb 03 2017

A370853 Numbers m such that c(0) < c(1) < c(2), where c(k) = number of k's in the ternary representation of m.

Original entry on oeis.org

17, 23, 25, 53, 71, 77, 79, 134, 152, 158, 160, 161, 206, 212, 214, 215, 230, 232, 233, 238, 239, 241, 296, 314, 320, 322, 350, 386, 398, 402, 404, 422, 428, 430, 440, 452, 456, 458, 464, 466, 470, 474, 476, 478, 480, 482, 484, 485, 530, 536, 538, 554, 556

Offset: 1

Clark Kimberling, Mar 03 2024

			The ternary representation of 17 is 122, for which c(0)=0 < c(1)=1 < c(2)=2.

John Tyler Rascoe, Table of n, a(n) for n = 1..10000

Cf. A007089, A077267, A062756, A081603.

Cf. A370854, A370855, A370856, A370859, A370863, A370870.

Select[Range[1000], DigitCount[#, 3, 0] < DigitCount[#, 3, 1] < DigitCount[#, 3, 2] &]

check(m) = {my(c0=0, c1=0, c2=0, s=Vec(digits(m, 3)));
for(i=1, length(s), if(s[i]==0, c0+=1, if(s[i]==1, c1+=1, if(s[i]==2, c2+=1,))));
c0John Tyler Rascoe, Mar 11 2024

cp's OEIS Frontend

A033016 Numbers whose base-3 expansion has no run of digits with length < 2.

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Mathematica

A037314 Numbers whose base-3 and base-9 expansions have the same digit sum.

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Julia

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A081605 Numbers having at least one 0 in their ternary representation.

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Haskell

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A118594 Palindromes in base 3 (written in base 3).

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A136808 Numbers k such that k and k^2 use only the digits 0, 1 and 2.

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Maple

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A265747 Numbers written in Jacobsthal greedy base.

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A062891 When expressed in base 3 and then interpreted in base 9, is a multiple of the original number.

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Maple