A005823 -id:A005823 - cp's OEIS frontend

A097251 Numbers whose set of base 5 digits is {0,4}.

0, 4, 20, 24, 100, 104, 120, 124, 500, 504, 520, 524, 600, 604, 620, 624, 2500, 2504, 2520, 2524, 2600, 2604, 2620, 2624, 3000, 3004, 3020, 3024, 3100, 3104, 3120, 3124, 12500, 12504, 12520, 12524, 12600, 12604, 12620, 12624, 13000, 13004, 13020

Offset: 0

Ray Chandler, Aug 03 2004

n such that there exists a permutation p_1, ..., p_n of 1, ..., n such that i + p_i is a power of 5 for every i.

The first 2^n terms of the sequence could be obtained using the Cantor-like process for the segment [0,5^n-1]. For example, for n=1 we have [0, {1,2,3},4] such that numbers outside of braces are the first 2 terms of the sequence; for n=2 we have [0, {1,2,3}, 4, {5,...,19}, 20, {21,22,23}, 24] such that the numbers outside of braces are the first 4 terms of the sequence, etc. - Vladimir Shevelev, Dec 17 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A001196, A005823, A097252-A097262.

[n: n in [0..20000] | Set(IntegerToSequence(n, 5)) subset {0, 4}]; // Vincenzo Librandi, May 25 2012

fQ[n_]:=Union@Join[{0,4},IntegerDigits[n,5]]=={0,4};Select[Range[0,20000],fQ] (* Vincenzo Librandi, May 25 2012 *)
FromDigits[#,5]&/@Tuples[{0,4},6] (* Harvey P. Dale, Feb 01 2015 *)

a[0]:0$ a[n]:=5*a[floor(n/2)]+2*(1-(-1)^n)$ makelist(a[n], n, 0, 42); /* Bruno Berselli, May 25 2012 */

a(n) = 4*fromdigits(binary(n),5); \\ Kevin Ryde, Jun 03 2020

a(n) = 4*A033042(n).

a(2n) = 5*a(n), a(2n+1) = a(2n)+4.

A121153 Numbers k with the property that 1/k can be written in base 3 in such a way that the fractional part contains no 1's.

Original entry on oeis.org

1, 3, 4, 9, 10, 12, 13, 27, 28, 30, 36, 39, 40, 81, 82, 84, 90, 91, 108, 117, 120, 121, 243, 244, 246, 252, 270, 273, 324, 328, 351, 360, 363, 364, 729, 730, 732, 738, 756, 757, 810, 819, 820, 949, 972, 984, 1036, 1053, 1080, 1089, 1092, 1093, 2187

Offset: 1

Jack W Grahl, Aug 12 2006

Numbers k such that 1/k is in the Cantor set.
A subsequence of A054591. The first member of A054591 which does not belong to this sequence is 146. See A135666.
This is not a subsequence of A005836 (949 belongs to the present sequence but not to A005836). See A170830, A170853.

			1/3 in base 3 can be written as either .1 or .0222222... The latter version contains no 1's, so 3 is in the sequence.
1/4 in base 3 is .02020202020..., so 4 is in the sequence.

T. D. Noe and N. J. A. Sloane, Table of n, a(n) for n=1..1164 (terms < 3^21)
D. Jordan and R. Schayer Rational points on the Cantor middle thirds set [Broken link corrected by _Rainer Rosenthal_, Feb 20 2009]

Cf. A054591, A005823, A005836, A170943, A170944, A170951, A170952, A170830, A170853.

(* Mma code from T. D. Noe, Feb 20 2010. This produces the sequence except for the powers of 3. *)
(* Find the length of the periodic part of the fraction: *)
FracLen[n_] := Module[{r = n/3^IntegerExponent[n, 3]}, MultiplicativeOrder[3, r]]
(* Generate the fractions and select those that have no 1's: *)
Select[Range[100000], ! MemberQ[Union[RealDigits[1/#, 3, FracLen[ # ]][[1]]], 1] &]

is(n,R=divrem(3^logint(n,3),n),S=0)={while(R[1]!=1&&!bittest(S,R[2]), S+=1<M. F. Hasler, Feb 27 2018

Extended to 10^5 by T. D. Noe and N. J. A. Sloane, Feb 20 2010

Entry revised by N. J. A. Sloane, Feb 22 2010

A055246 At step number k >= 1 the 2^(k-1) open intervals that are erased from [0,1] in the Cantor middle-third set construction are I(k,n) = (a(n)/3^k, (1+a(n))/3^k), n=1..2^(k-1).

Original entry on oeis.org

1, 7, 19, 25, 55, 61, 73, 79, 163, 169, 181, 187, 217, 223, 235, 241, 487, 493, 505, 511, 541, 547, 559, 565, 649, 655, 667, 673, 703, 709, 721, 727, 1459, 1465, 1477, 1483, 1513, 1519, 1531, 1537, 1621, 1627, 1639, 1645, 1675, 1681, 1693, 1699

Offset: 1

Wolfdieter Lang, May 23 2000

Related to A005836. Gives boundaries of open intervals that have to be erased in the Cantor middle-third set construction.
Let g(n) = Sum_{i=0..n} (i*binomial(n+i,i)^3*binomial(n,i)^2) = A112035(n). Let b = {m>0 : g(m) != 0 (mod 3)}. Then b(n) = a(n). - Mohammed Bouayoun (bouyao(AT)wanadoo.fr), Mar 08 2004
Conjecture: Similarly to A191107, this increasing sequence is generated by the rules: a(1) = 1, and if x is in the sequence, then 3*x-2 and 3*x+4 are also in the sequence. - L. Edson Jeffery, Nov 17 2015

			k=1: (1/3, 2/3);
k=2: (1/9, 2/9), (7/9, 8/9);
k=3: (1/27, 2/27), (7/27, 8/27), (19/27, 20/27), (25/27, 26/27); ...

Ralf Stephan, Some divide-and-conquer sequences ...
Ralf Stephan, Table of generating functions
Index entries for 3-automatic sequences.

Cf. A003278, A005836, A005823, A055247, A112035.

Cf. A191107.

(* (Conjectured) Choose rows large enough to guarantee that all terms < max are generated. *)
rows = 1000; max = 10^4; a[1] = {1}; i = 1; Do[a[n_] = {}; Do[If[1 < 3*a[n - 1][[k]] - 2 < max, AppendTo[a[n], 3*a[n - 1][[k]] - 2], Break]; If[3*a[n - 1][[k]] + 4 < max, AppendTo[a[n], 3*a[n - 1][[k]] + 4], Break], {k, Length[a[n - 1]]}]; If[a[n] == {}, Break, i++], {n, 2, 1000}]; a055246 = Take[Flatten[Table[a[n], {n, i}]], 48] (* L. Edson Jeffery, Nov 17 2015 *)
Join[{1}, 1 + 6 Accumulate[Table[(3^IntegerExponent[n, 2] + 1)/2, {n, 60}]]] (* Vincenzo Librandi, Nov 26 2015 *)

g(n)=sum(i=0,n,i*binomial(n+i,i)^3*binomial(n,i)^2);
for (i=1,2000,if(Mod(g(i),3)<>0,print1(i,",")))

a(n) = fromdigits(binary(n-1),3)*6 + 1; \\ Kevin Ryde, Apr 23 2021

def A055246(n): return int(bin(n-1)[2:],3)*6|1 # Chai Wah Wu, Jun 26 2025

a(n) = 1+6*A005836(n), n >= 1.
a(n) = 1+3*A005823(n), n >= 1.
a(n+1) = A074938(n) + A074939(n); A074938: odd numbers in A005836, A074939: even numbers in A005836. - Philippe Deléham, Jul 10 2005
Conjecture: a(n) = 2*A191107(n) - 1 = 6*A003278(n) - 5 = (a((2*n-1)*2^(k-1))+2)/3^k, k>0. - L. Edson Jeffery, Nov 25 2015

Edited by N. J. A. Sloane, Nov 20 2015: used first comment to give more precise definition, and edited a comment at the suggestion of L. Edson Jeffery.

A081606 Numbers having at least one 1 in their ternary representation.

Original entry on oeis.org

1, 3, 4, 5, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 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, 81, 82, 83, 84, 85, 86

Offset: 1

Reinhard Zumkeller, Mar 23 2003

Complement of A005823.
Integers m such that central Delannoy number A001850(m) == 0 (mod 3). - Emeric Deutsch and Bruce E. Sagan, Dec 04 2003
Integers m such that A026375(m) == 0 (mod 3). - Fabio Visonà, Aug 03 2023

Johann Cigler, Some elementary observations on Narayana polynomials and related topics, arXiv:1611.05252 [math.CO], 2016. See p.25.
Mathematics Stack Exchange, Proof that integers m in this sequence are such that A026375(m) == 0 (mod 3).

Cf. A007089, A062756, A081609, A081605, A074940.

Select[Range[100],DigitCount[#,3,1]>0&] (* Harvey P. Dale, Nov 26 2022 *)

from itertools import count, islice
def A081606_gen(): # generator of terms
    a = 0
    for n in count(1):
        b = int(bin(n)[2:],3)<<1
        yield from range(a+1,b)
        a = b
A081606_list = list(islice(A081606_gen(),30)) # Chai Wah Wu, Oct 13 2023

from gmpy2 import digits
def A081606(n):
    def f(x):
        s = digits(x>>1,3)
        for i in range(l:=len(s)):
            if s[i]>'1':
                break
        else:
            return n+int(s,2)
        return n-1+(int(s[:i] or '0',2)+1<Chai Wah Wu, Oct 29 2024

More terms from Emeric Deutsch and Bruce E. Sagan, Dec 04 2003

A250047 Numbers n such that m = floor(n/7) is not coprime to n and, if nonzero, m is also a term of the sequence.

Original entry on oeis.org

2, 3, 4, 5, 6, 14, 16, 18, 20, 21, 24, 27, 28, 30, 32, 34, 35, 40, 42, 44, 45, 46, 48, 98, 100, 102, 104, 112, 114, 116, 118, 126, 128, 129, 130, 132, 140, 142, 144, 145, 146, 147, 150, 153, 168, 170, 171, 172, 174, 189, 192, 195, 196, 198, 200, 202, 210

Offset: 1

Stanislav Sykora, Jan 15 2015

See the comments in A250041 which all apply, except for the setting of the base, b=7. In particular, they define the property RTNC(b).

Stanislav Sykora, Table of n, a(n) for n = 1..10000
Stanislav Sykora, PARI/GP scripts for genetic threads, with code and comments
Wikipedia, Coprime integers

Cf. A250041 (b=10), A250046.
Other lists of right-truncatable numbers with the property RTNC(b):
A005823 (b=3), A250037 (b=4), A250039 (b=16), A250043 (b=9), A250045 (b=8), A250049 (b=6), A250051 (b=5).

PARI
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See the link
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is_rtnc(n, b=7) = {while (((m=gcd(n\b, n)) != 1), if (m == 0, return (1)); n = n\b; ); return (0); } \\ Michel Marcus, Jan 22 2015

A088917 Central Delannoy numbers (mod 3); Characteristic function for Cantor set.

Original entry on oeis.org

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

Offset: 0

Benoit Cloitre, Nov 30 2003

nonn

Also Apery numbers (mod 3).
More generally also (Sum_{k=0..n} binomial(n,k)^x*binomial(n+k,k)^y) (mod 3) for any x >= 1 in N and any odd y >= 1.
a(n) = 0 if the ternary expansion of n contains one or more 1-digits, otherwise 1. - Antti Karttunen, Aug 23 2019
Main diagonal of the Sierpinski carpet (A153490). - Paolo Xausa, May 19 2023

Antti Karttunen, Table of n, a(n) for n = 0..59048
Michael Coons and James Evans, A sequential view of self-similar measures, or, What the ghosts of Mahler and Cantor can teach us about dimension, arXiv:2011.10722 [math.NT], 2020. See Figure 2 p. 2.
Eric Weisstein's World of Mathematics, Cantor Fractal.
Index entries for characteristic functions

Cf. A005258, A081606, A001850, A105220, A153490, A316829.

Characteristic function of A005823, and with offset 1, characteristic function of A191106.

Nest[ Flatten[# /. {0 -> {0, 0, 0}, 1 -> {1, 0, 1}}] &, {1}, 5] (* Or *)
f[n_] := Mod[LegendreP[n, 3], 3]; Array[f, 111, 0] (* Or *)
f[n_] := If[ FreeQ[ IntegerDigits[n, 3], 1], 1, 0]; Array[f, 111, 0] (* also from Mathematica v8.0 Mathematical Functions Help section for "IntegerDigits" "Cantor set construction:" *) (* Robert G. Wilson v, Jun 16 2011 *)
Nest[Join[#, 0 #, #] &, {1}, 5] (* IWABUCHI Yu(u)ki, Sep 08 2012 *)

a(n)=sum(k=0,n,binomial(n,k)*binomial(n+k,k))%3

A088917(n) = { while(n, if(n%3==1, return(0), n\=3)); (1); }; \\ Antti Karttunen, Aug 23 2019 (copied from A005823)

A088917(n) = abs(factorback(apply(d -> d-1,digits(n,3)))); \\ Antti Karttunen, Aug 23 2019

a(A005823(n)) = 1; a(A081606(n)) = 0.
a(n) = A001850(n) - 3*floor(A001850(n)/3).
a(n) = 2 - A105220(n) = 1 - A316829(n). - Antti Karttunen and Jon Maiga, Aug 24 2019
G.f.: Product_{k>=0} (1 + x^(2*3^k)). - Ilya Gutkovskiy, Jun 05 2021

Secondary name added by Antti Karttunen, Aug 23 2019

A097256 Numbers whose set of base 10 digits is {0,9}.

Original entry on oeis.org

0, 9, 90, 99, 900, 909, 990, 999, 9000, 9009, 9090, 9099, 9900, 9909, 9990, 9999, 90000, 90009, 90090, 90099, 90900, 90909, 90990, 90999, 99000, 99009, 99090, 99099, 99900, 99909, 99990, 99999, 900000, 900009, 900090, 900099, 900900

Offset: 0

Ray Chandler, Aug 03 2004

n such that there exists a permutation p_1, ..., p_n of 1, ..., n such that i + p_i is a power of 10 for every i.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A001196, A005823, A097251-A097262.

Cf. A078248, A169965, A169966, A169967, A169964, A204093, A204094, A204095.

a097256 n = a097256_list !! n
a097256_list = map (* 9) a007088_list
-- Reinhard Zumkeller, Jan 10 2012

A097256:=n->(9/2) * add((1-(-1)^floor(n/2^i))*10^i, i=0..n); seq(A097256(n), n=0..30); # Wesley Ivan Hurt, Feb 11 2014

Table[(9/2) Sum[(1 - (-1)^Floor[n/2^i]) 10^i, {i, 0, n}], {n, 0, 30}] (* Wesley Ivan Hurt, Feb 11 2014 *)

a(n) = 9*A007088(n).

a(2n) = 10*a(n), a(2n+1) = a(2n)+9.

A191108 Increasing sequence generated by these rules: a(1)=1, and if x is in a then 3x-2 and 3x+2 are in a.

Original entry on oeis.org

1, 5, 13, 17, 37, 41, 49, 53, 109, 113, 121, 125, 145, 149, 157, 161, 325, 329, 337, 341, 361, 365, 373, 377, 433, 437, 445, 449, 469, 473, 481, 485, 973, 977, 985, 989, 1009, 1013, 1021, 1025, 1081, 1085, 1093, 1097, 1117, 1121, 1129, 1133, 1297, 1301, 1309, 1313, 1333, 1337, 1345, 1349, 1405, 1409, 1417, 1421, 1441, 1445

Offset: 1

Clark Kimberling, May 26 2011

See discussions at A190803, A191106.  The sequence a=A191108 has closure properties:  the positive integers in (2+A191108)/3 comprise A191108, as do those in (-2+A191108)/3.
From Peter Munn, May 13 2019: (Start)
The closure of {1} in the positive integers under reflection about 3^k, k >= 1.
Asymptotic density is 0.
Consider a Sierpinski arrowhead curve formed of edges numbered consecutively from 0 at its axis of symmetry. The m-th edge is contained in the boundary of the plane sector occupied by the arrowhead if and only if m or -m is in this sequence.
For k >= 0, a(2^k) = 2*3^k - 1 and {a(i)/(2*3^k) | 1 <= i <= 2^k} is the set of center points of surviving intervals at the k-th step of generating the Cantor set, and therefore the set of center points of deleted middle-third intervals at the (k+1)-th step.
Define t: Z -> P(R) so that t(n) is the translated Cantor ternary set spanning [(n-1)/2, (n+1)/2], 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.
(End)

Ivan Neretin, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Cantor Set
Wikipedia, Sierpiński arrowhead curve

Cf. A005823, A005836, A055247, A147991, A190803, A191106, A306556.

h = 3; i = -2; j = 3; k = 2; f = 1; g = 7;
a = Union[Flatten[NestList[{h # + i, j # + k} &, f, g]]]  (* A191108 *)
b = (a + 2)/3; c = (a - 2)/3; r = Range[1, 900];
d = Intersection[b, r] (* A191108 closure property  *)
e = Intersection[c, r] (* A191108 closure property  *)

a(n) = fromdigits(binary(n-1),3)<<2 + 1; \\ Kevin Ryde, Aug 05 2022

From Peter Munn, May 25 2019: (Start)
a(n) = (A055247(2n-1) + A055247(2n)) / 3.
a(n) = A306556(2n)*2 - 1 = A306556(2n-1) + A306556(2n).
a(n) = 2*A005823(n) + 1 = 4*A005836(n) + 1 = 2*A191106(n) - 1.
a(2^k+i) = 2*A147991(2^k+i-1) + 3^(k+1) for k >= 0, 1 <= i <= 2^k.
(End)

A250037 Numbers n such that m = floor(n/4) is not coprime to n and, if nonzero, m is also a term of the sequence.

Original entry on oeis.org

2, 3, 8, 10, 12, 15, 32, 34, 40, 42, 48, 50, 51, 60, 63, 128, 130, 136, 138, 160, 162, 168, 170, 171, 192, 194, 195, 200, 202, 204, 207, 240, 242, 243, 252, 255, 512, 514, 520, 522, 544, 546, 552, 554, 555, 640, 642, 648, 650, 651, 672, 674, 675, 680, 682

Offset: 1

Stanislav Sykora, Dec 07 2014

See the comments in A250041 which all apply, except for the setting of the base, b=4. In particular, they define the property RTNC(b).

If x =12*k+j, 0 <= j <= 11, then x is in the sequence iff either j is in {0,2,3} and 3*k is in the sequence, or j is in {4,6} and 3*k+1 is in the sequence, or j is in {8,10} and 3*k+2 is in the sequence. - Robert Israel, Dec 22 2014

Stanislav Sykora, Table of n, a(n) for n = 1..10000
Stanislav Sykora, PARI/GP scripts for genetic threads, with code and comments.
Wikipedia, Coprime integers

Cf. A250036, A250041.
Other lists of right-truncatable numbers with the property RTNC(b):
A005823 (b=3), A250039 (b=16), A250041 (b=10), A250043 (b=9), A250045 (b=8), A250047 (b=7), A250049 (b=6), A250051 (b=5).

S:= {}:
for n from 1 to 1000 do
  m:= floor(n/4);
  if igcd(m,n) = 1 then next fi;
  if m > 0 and not member(m,S) then next fi;
  S:= S union {n}
od:
S; # if using Maple 11 or earlier, uncomment the next line
# sort(convert(S,list)); # Robert Israel, Dec 22 2014

PARI
```
See the link.
```

is_rtnc(n, b=4) = {while (((m=gcd(n\b, n)) != 1), if (m == 0, return (1)); n = n\b; ); return (0); } \\ Michel Marcus, Jan 22 2015

A250039 Numbers n such that m = floor(n/16) is not coprime to n and, if nonzero, m is also a term of the sequence.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 32, 34, 36, 38, 40, 42, 44, 46, 48, 51, 54, 57, 60, 63, 64, 66, 68, 70, 72, 74, 76, 78, 80, 85, 90, 95, 96, 98, 99, 100, 102, 104, 105, 106, 108, 110, 111, 112, 119, 126, 128, 130, 132, 134, 136, 138, 140, 142

Offset: 1

Stanislav Sykora, Dec 07 2014

See the comments in A250041 which all apply, except for the setting of the base, b=16. In particular, they define the property RTNC(b).

This list is an infinite subset of A248502 with which it shares the first 111 entries.

Stanislav Sykora, Table of n, a(n) for n = 1..10000
Stanislav Sykora, PARI/GP scripts for genetic threads, with code and comments.
Wikipedia, Coprime integers

Cf. A248502, A250038, A250041.

Other lists of right-truncatable numbers with the property RTNC(b): A005823 (b=3), A250037 (b=4), A250041 (b=10), A250043 (b=9), A250045 (b=8), A250047 (b=7), A250049 (b=6), A250051 (b=5).

PARI
```
See the link.
```

is_rtnc(n, b=16) = {while (((m=gcd(n\b, n)) != 1), if (m == 0, return (1)); n = n\b; ); return (0); } \\ Michel Marcus, Jan 29 2015

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A097251 Numbers whose set of base 5 digits is {0,4}.

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A121153 Numbers k with the property that 1/k can be written in base 3 in such a way that the fractional part contains no 1's.

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A055246 At step number k >= 1 the 2^(k-1) open intervals that are erased from [0,1] in the Cantor middle-third set construction are I(k,n) = (a(n)/3^k, (1+a(n))/3^k), n=1..2^(k-1).

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A081606 Numbers having at least one 1 in their ternary representation.

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A250047 Numbers n such that m = floor(n/7) is not coprime to n and, if nonzero, m is also a term of the sequence.

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A088917 Central Delannoy numbers (mod 3); Characteristic function for Cantor set.

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A097256 Numbers whose set of base 10 digits is {0,9}.

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