A005823 -id:A005823 - cp's OEIS frontend

A250041 Numbers n such that m = floor(n/10) is not coprime to n and, if nonzero, m is also a term of the sequence.

2, 3, 4, 5, 6, 7, 8, 9, 20, 22, 24, 26, 28, 30, 33, 36, 39, 40, 42, 44, 46, 48, 50, 55, 60, 62, 63, 64, 66, 68, 69, 70, 77, 80, 82, 84, 86, 88, 90, 93, 96, 99, 200, 202, 204, 205, 206, 208, 220, 222, 224, 226, 228, 240, 242, 243, 244, 246, 248, 249, 260, 262

Offset: 1

Stanislav Sykora, Dec 07 2014

Equivalent definition 1: Assuming a base b (in this case b=10), let us say that a positive integer k has the property RTNC(b) when m=floor(k/b) is not coprime to k, i.e., gcd(k,m)>1. Then k belongs to this sorted list if (i) it has the property RTNC(b) and (ii) m is either 0 or belongs also to the list.
Equivalent definition 2: Every nonempty prefix of a(n) in base b has the property RTNC(b).
Notes: The acronym RTNC stands for 'Right-Truncated is Not Coprime' (negation of the property RTC defined in A250040). We could also say that a(n) are right-truncatable numbers with property RTNC(b).
This particular list is an infinite subset of A248500.

			243 is a member because (243,24), (24,2) and (2,0) are noncoprime pairs.
155 is not a member because gcd(15,1)=1.

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. A248500, A250040.

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), A250047 (b=7), A250049 (b=6), A250051 (b=5).

PARI
```
See the link.
```

is_rtnc(n, b=10) = {while (((m=gcd(n\b, n)) != 1), if (m == 0, return (1)); n = n\b; ); return (0); } \\ Michel Marcus, Jan 29 2015

A250043 Numbers n such that m = floor(n/9) 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, 18, 20, 22, 24, 26, 27, 30, 33, 36, 38, 40, 42, 44, 45, 50, 54, 56, 57, 58, 60, 62, 63, 70, 72, 74, 76, 78, 80, 162, 164, 165, 166, 168, 170, 180, 182, 184, 185, 186, 188, 198, 200, 202, 204, 206, 216, 218, 219, 220, 222, 224, 234, 236

Offset: 1

Stanislav Sykora, Jan 15 2015

See the comments in A250041 which all apply, except for the setting of the base, b=9. 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), A250042.

Other lists of right-truncatable numbers with the property RTNC(b): A005823 (b=3), A250037 (b=4), A250039 (b=16), A250045 (b=8), A250047 (b=7), A250049 (b=6), A250051 (b=5).

PARI
```
See the link
```

is_rtnc(n, b=9) =  {while (((m=gcd(n\b, n)) != 1), if (m == 0, return (1)); n = n\b;); return (0);} \\ Michel Marcus, Jan 17 2015

A250045 Numbers n such that m = floor(n/8) 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, 16, 18, 20, 22, 24, 27, 30, 32, 34, 36, 38, 40, 45, 48, 50, 51, 52, 54, 56, 63, 128, 130, 132, 134, 144, 146, 147, 148, 150, 160, 162, 164, 165, 166, 176, 178, 180, 182, 192, 194, 195, 196, 198, 216, 219, 222, 240, 242, 243, 244, 245, 246

Offset: 1

Stanislav Sykora, Jan 15 2015

See the comments in A250041 which all apply, except for the setting of the base, b=8. 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), A250044.
Other lists of right-truncatable numbers with the property RTNC(b):
A005823 (b=3), A250037 (b=4), A250039 (b=16), A250043 (b=9), A250047 (b=7), A250049 (b=6), A250051 (b=5).

PARI
```
See the link
```

is_rtnc(n, b=8) = {while (((m=gcd(n\b, n)) != 1), if (m == 0, return (1)); n = n\b; ); return (0); } \\ Michel Marcus, Jan 22 2015

A154314 Numbers with not more than two distinct digits in ternary representation.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 16, 17, 18, 20, 22, 23, 24, 25, 26, 27, 28, 30, 31, 36, 37, 39, 40, 41, 43, 44, 49, 50, 52, 53, 54, 56, 60, 62, 67, 68, 70, 71, 72, 74, 76, 77, 78, 79, 80, 81, 82, 84, 85, 90, 91, 93, 94, 108, 109, 111, 112, 117, 118, 120, 121, 122

Offset: 1

Reinhard Zumkeller, Jan 07 2009

Reinhard Zumkeller, 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.

Complement of A031944.
Union of A032924, A005823 and A005836.
Cf. A007089, A043530, A212193.

import Data.List (findIndices)
a154314 n = a154314_list !! (n-1)
a154314_list = findIndices (/= 3) a212193_list
-- Reinhard Zumkeller, May 04 2012

Select[Range[0,200],Length[Union[IntegerDigits[#,3]]]<3&] (* Harvey P. Dale, Nov 23 2012 *)

is(n)=#Set(digits(n,3))<3 \\ Charles R Greathouse IV, Mar 17 2014

A043530(a(n)) <= 2.
A212193(a(n)) <> 3. - Reinhard Zumkeller, May 04 2012
a(n) >> n^1.58..., where the exponent is log(3)/log(2). - Charles R Greathouse IV, Mar 17 2014
Sum_{n>=2} 1/a(n) = 5.47555542241781419692840472181029603722178623821762258873485212626135391726959422416350447132335696748507... (calculated using Baillie and Schmelzer's kempnerSums.nb, see Links). - Amiram Eldar, Apr 14 2025

A250049 Numbers n such that m = floor(n/6) 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, 12, 14, 16, 18, 21, 24, 26, 28, 30, 35, 72, 74, 75, 76, 84, 86, 88, 96, 98, 100, 108, 110, 111, 112, 126, 129, 144, 146, 147, 148, 156, 158, 160, 168, 170, 172, 180, 182, 183, 184, 185, 210, 215, 432, 434, 435, 436, 444, 446, 448, 450, 453, 455

Offset: 1

Stanislav Sykora, Jan 31 2015

See the comments in A250041 which all apply, except for the setting of the base, b=6. 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), A250047 (b=7), A250051 (b=5).

PARI
```
\\ See A250041 for b=6
```

is_rtnc(n, b=6) = {while (((m=gcd(n\b, n)) != 1), if (m == 0, return (1)); n = n\b;); return (0); } \\ Michel Marcus, Jan 31 2015

A250051 Numbers n such that m = floor(n/5) is not coprime to n and, if nonzero, m is also a term of the sequence.

Original entry on oeis.org

2, 3, 4, 10, 12, 14, 15, 18, 20, 22, 24, 50, 52, 54, 60, 62, 63, 64, 70, 72, 74, 75, 78, 90, 92, 93, 94, 100, 102, 104, 110, 112, 114, 120, 122, 123, 124, 250, 252, 254, 260, 262, 264, 270, 272, 273, 274, 300, 302, 303, 304, 310, 312, 314, 315, 318

Offset: 1

Stanislav Sykora, Jan 31 2015

See the comments in A250041 which all apply, except for the setting of the base, b=5. 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), A250047 (b=7), A250049 (b=6).

PARI
```
\\ See A250041 for b=5
```

is_rtnc(n, b=5) = {while (((m=gcd(n\b, n)) != 1), if (m == 0, return (1)); n = n\b;); return (0);} \\ Michel Marcus, Jan 31 2015

A125292 Numbers having either no ones or no twos in their ternary representation.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 10, 12, 13, 18, 20, 24, 26, 27, 28, 30, 31, 36, 37, 39, 40, 54, 56, 60, 62, 72, 74, 78, 80, 81, 82, 84, 85, 90, 91, 93, 94, 108, 109, 111, 112, 117, 118, 120, 121, 162, 164, 168, 170, 180, 182, 186, 188, 216, 218, 222, 224, 234, 236, 240, 242, 243

Offset: 1

Reinhard Zumkeller, Nov 26 2006

Complement of A125293; union of A005823 and A005836;

A125291(a(n)) = 1; A062756(a(n))*A081603(a(n)) = 0.

Rémy Sigrist, Table of n, a(n) for n = 1..8190
Eric Weisstein's World of Mathematics, Ternary
Eric Weisstein's World of Mathematics, Repdigit

Subsequence of A154314.

Cf. A007089, A125289.

not[n_]:=Module[{c=DigitCount[n,3]},c[[1]]==0||c[[2]]==0]; Select[ Range[ 250],not] (* Harvey P. Dale, Dec 15 2012 *)

is(n, base=3) = #Set(select(sign, digits(n, base)))==1 \\ Rémy Sigrist, Mar 28 2020

a(n, base=3) = { 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

A306556 Integers that appear as (unreduced) numerators of segment endpoints when a ternary Cantor set is created.

Original entry on oeis.org

0, 1, 2, 3, 6, 7, 8, 9, 18, 19, 20, 21, 24, 25, 26, 27, 54, 55, 56, 57, 60, 61, 62, 63, 72, 73, 74, 75, 78, 79, 80, 81, 162, 163, 164, 165, 168, 169, 170, 171, 180, 181, 182, 183, 186, 187, 188, 189, 216, 217, 218, 219, 222, 223, 224, 225, 234, 235, 236, 237, 240, 241, 242, 243

Offset: 1

Dan Dima, Feb 23 2019

Nonnegative integers whose ternary representation contains only digits 0 and 2 except for at most a single digit 1 that is followed only by 0's.
Nonnegative integers that can be written in base 3 using only 0's and 2's, allowing the use of the "decimal" point (.) and replacing ....10..0(.) by ....02..2(.)2222...
Note that fractions are not reduced.
List of integers in the closure of the ternary Cantor set under multiplication by 3. The closure is the union of the translated ternary Cantor sets spanning [a(1), a(2)], [a(3), a(4)], [a(5), a(6)], ... . - Peter Munn, Jul 09 2019

			On 1st step we have [0,1/3] U [2/3,3/3] so we get a(1)=0, a(2)=1, a(3)=2, a(4)=3.
On 2nd step we have [0,1/9] U [2/9,3/9] U [6/9,7/9] U [8/9,9/9] so we get in addition a(5)=6, a(6)=7, a(7)=8, a(8)=9.

Georg Cantor, Über unendliche, lineare Punktmannigfaltigkeiten V" [On infinite, linear point-manifolds (sets), Part 5]. Mathematische Annalen (in German). (1883) 21: 545-591.
Paul du Bois-Reymond, Der Beweis des Fundamentalsatzes der Integralrechnung, Mathematische Annalen (in German), (1880), 16, footnote on p. 128.
Eric Weisstein's World of Mathematics, Cantor Set
Wikipedia, Cantor set
Index entries for 3-automatic sequences.

Cf. A005823, A054591, A055247, A191106, A191108.

A306556(n) = {sm=0;while(n>1,ex=floor(log(n)/log(2));if(n-2^ex==0,sm=sm+3^(ex-1),sm=sm+2*3^(ex-1));n=n-2^ex);return(sm)}

a(n) = n--; fromdigits(binary(n>>1),3)*2 + (n%2); \\ Kevin Ryde, Apr 23 2021

a(1)=0, a(2)=1;
a(2^n) = 3^(n-1) for n >= 1;
a(2^n+k) = 2*3^(n-1) + a(k) for 1 <= k <= 2^n.
From Peter Munn, Jul 09 2019: (Start)
a(2n-1) = A005823(n) = A191106(n)-1.
a(2n)   = A191106(n) = A005823(n)+1.
a(2n-1) = (A055247(2n-1)-1)/3.
a(2n)   = (A055247(2n)  +1)/3.
a(2n-1) = (A191108(n)-1)/2.
a(2n)   = (A191108(n)+1)/2.
(End)

A328849 Numbers in whose primorial base expansion only even digits appear.

Original entry on oeis.org

0, 4, 12, 16, 24, 28, 60, 64, 72, 76, 84, 88, 120, 124, 132, 136, 144, 148, 180, 184, 192, 196, 204, 208, 420, 424, 432, 436, 444, 448, 480, 484, 492, 496, 504, 508, 540, 544, 552, 556, 564, 568, 600, 604, 612, 616, 624, 628, 840, 844, 852, 856, 864, 868, 900, 904, 912, 916, 924, 928, 960, 964, 972, 976, 984, 988, 1020, 1024

Offset: 1

Antti Karttunen, Oct 30 2019

Numbers for which the prime factor form (A276086) of their primorial base expansion is a square, A000290.

			144 is written as "4400" in primorial base (A049345), because 4*A002110(3) + 4*A002110(2) + 0*A002110(1) + 0*A002110(0) = 4*30 + 4*6 = 144, thus all the digits are even and 144 is included in this sequence.

Antti Karttunen, Table of n, a(n) for n = 1..9072
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..90720
Index entries for sequences related to primorial base.

Cf. A000290, A002110, A010052, A049345, A276086, A276156, A328770.
Cf. A328834, A328850 (squares in this sequence).
Similar sequences: A005823 (ternary), A014263 (decimal), A062880 (quaternary), A351893 (factorial base).

With[{max = 5}, bases = Prime@ Range[max, 1, -1]; nmax = Times @@ bases - 1; prmBaseDigits[n_] := IntegerDigits[n, MixedRadix[bases]]; Select[Range[0, nmax, 2], AllTrue[prmBaseDigits[#], EvenQ] &]] (* Amiram Eldar, May 23 2023 *)

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
isA328849(n) = issquare(A276086(n));

a(n) = 2*A328770(n).

A000196(A276086(a(n))) = A276086(a(n)/2) = A328834(n).

A097252 Numbers whose set of base 6 digits is {0,5}.

Original entry on oeis.org

0, 5, 30, 35, 180, 185, 210, 215, 1080, 1085, 1110, 1115, 1260, 1265, 1290, 1295, 6480, 6485, 6510, 6515, 6660, 6665, 6690, 6695, 7560, 7565, 7590, 7595, 7740, 7745, 7770, 7775, 38880, 38885, 38910, 38915, 39060, 39065, 39090, 39095, 39960, 39965

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 6 for every i.

Vincenzo Librandi, Table of n, a(n) for n = 0..500

Cf. A001196, A005823, A007088, A033043, A097251-A097262.

[n: n in [0..40000] | Set(IntegerToSequence(n, 6)) subset {0, 5}]; // Vincenzo Librandi, May 25 2012

fQ[n_]:=Union@Join[{0,5},IntegerDigits[n,6]]=={0,5};Select[Range[0,40000],fQ] (* Vincenzo Librandi, May 25 2012 *)
FromDigits[#,6]&/@Tuples[{ 0,5},6] (* Harvey P. Dale, Aug 15 2021 *)

def A079252(n): return 5*int(bin(n)[2:],6) # Chai Wah Wu, Apr 04 2025

a(n) = 5*A033043(n).

a(2n) = 6*a(n), a(2n+1) = a(2n)+5.

cp's OEIS Frontend

A250041 Numbers n such that m = floor(n/10) is not coprime to n and, if nonzero, m is also a term of the sequence.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

PARI

A250043 Numbers n such that m = floor(n/9) is not coprime to n and, if nonzero, m is also a term of the sequence.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

PARI

A250045 Numbers n such that m = floor(n/8) is not coprime to n and, if nonzero, m is also a term of the sequence.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

PARI

A154314 Numbers with not more than two distinct digits in ternary representation.

Views

Author

Keywords

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Formula

A250049 Numbers n such that m = floor(n/6) is not coprime to n and, if nonzero, m is also a term of the sequence.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

PARI

A250051 Numbers n such that m = floor(n/5) is not coprime to n and, if nonzero, m is also a term of the sequence.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

PARI

A125292 Numbers having either no ones or no twos in their ternary representation.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

PARI

A306556 Integers that appear as (unreduced) numerators of segment endpoints when a ternary Cantor set is created.

Views

Author

Keywords

Comments

Examples