A031944 -id:A031944 - cp's OEIS frontend

A049354 Digitally balanced numbers in base 3: equal numbers of 0's, 1's, 2's.

11, 15, 19, 21, 260, 266, 268, 278, 290, 294, 302, 304, 308, 312, 316, 318, 332, 344, 348, 380, 384, 396, 410, 412, 416, 420, 424, 426, 434, 438, 450, 460, 462, 468, 500, 502, 508, 518, 520, 524, 528, 532, 534, 544, 550, 552, 572, 574, 578, 582, 586, 588, 596

Offset: 1

Harvey P. Dale

Alois P. Heinz, Table of n, a(n) for n = 1..2000

Cf. A031443, A031944.
Cf. A049354-A049360. See also A061854, A037861.
Cf. A062756, A077267, A081603.
Row n = 3 of A378000.

a049354 n = a049354_list !! (n-1)
a049354_list = filter f [1..] where
   f n = t0 == a062756 n && t0 == a081603 n where t0 = a077267 n
-- Reinhard Zumkeller, Aug 09 2014

Select[Range[600],Length[Union[DigitCount[#,3]]]== 1&]
FromDigits[#,3]&/@DeleteCases[Flatten[Permutations/@Table[PadRight[{},3n,{1,0,2}],{n,3}],1],?(#[[1]]==0&)]//Sort (* _Harvey P. Dale, May 30 2016 *)
Select[Range@5000, Differences@DigitCount[#,3]=={0,0}&] (* Hans Rudolf Widmer, Dec 11 2021 *)

from sympy.ntheory import count_digits
def ok(n): c = count_digits(n, 3); return c[0] == c[1] == c[2]
print([k for k in range(600) if ok(k)]) # Michael S. Branicky, Nov 15 2021

A062756(a(n)) = A077267(a(n)) and A081603(a(n)) = A077267(a(n)). - Reinhard Zumkeller, Aug 09 2014

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

A212193 In ternary representation of n: a(n) = if n is pandigital then 3 else least digit not used.

Original entry on oeis.org

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

Offset: 0

Reinhard Zumkeller, May 04 2012

a(A032924(n)) = 0; a(A081605(n)) <> 0;

a(A031944(n)) = 3; a(A154314(n)) <> 3.

			.   0 ->   '0':   a(0) = 1
.   1 ->   '1':   a(1) = 0
.   2 ->   '2':   a(2) = 0
.   3 ->  '10':   a(3) = 2
.   4 ->  '11':   a(4) = 0
.   5 ->  '12':   a(5) = 0
.   6 ->  '20':   a(6) = 1
.   7 ->  '21':   a(7) = 0
.   8 ->  '22':   a(8) = 0
.   9 -> '100':   a(9) = 2
.  10 -> '101':  a(10) = 2
.  11 -> '102':  a(11) = 3  <-- 11 is the smallest 3-pandigital number
.  12 -> '110':  a(12) = 2
.  13 -> '111':  a(13) = 0
.  14 -> '112':  a(14) = 0
.  15 -> '120':  a(15) = 3.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Ternary
Eric Weisstein's World of Mathematics, Pandigital Number
Wikipedia, Ternary numeral system
Wikipedia, Pandigital number

Cf. A007089, A067898 (decimal).

import Data.List (delete)
a212193 n = f n [0..3] where
   f x ys | x <= 2    = head $ delete x ys
          | otherwise = f x' $ delete d ys where (x',d) = divMod x 3

A043530 Number of distinct base-3 digits of n.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 2, 1, 2, 2, 3, 2, 1, 2, 3, 2, 2, 2, 3, 2, 3, 2, 2, 2, 2, 1, 2, 2, 3, 2, 2, 3, 3, 3, 3, 2, 2, 3, 2, 1, 2, 3, 2, 2, 3, 3, 3, 3, 2, 2, 3, 2, 2, 2, 3, 2, 3, 3, 3, 2, 3, 2, 3, 3, 3, 3, 2, 2, 3, 2, 2, 2, 3, 2, 3, 2, 2, 2, 2, 1, 2, 2, 3, 2, 2, 3, 3, 3, 3, 2

Offset: 1

Clark Kimberling

a(A031944(n)) = 3; a(A154314(n)) < 3. - Reinhard Zumkeller, Jan 07 2009

Image, under the coding sending i -> 1+floor(i/3), of the fixed point, starting with 0, of the morphism 0 -> 012, 1 -> 314, 2 -> 542, 3 -> 336, 4 -> 644, 5 -> 565, 6 -> 666. - Jeffrey Shallit, May 15 2016

Antti Karttunen, Table of n, a(n) for n = 1..19683

Cf. A007089, A031944, A037897, A154314.

Table[Length[Union[IntegerDigits[n,3]]],{n,90}] (* Harvey P. Dale, May 13 2020 *)

a(n) = #vecsort(digits(n,3),,8); \\ Michel Marcus, May 16 2016

A158928 a(n) is the smallest integer not yet in the sequence with no common base-3 digit with a(n-1).

Original entry on oeis.org

1, 2, 3, 8, 4, 6, 13, 18, 40, 20, 121, 24, 364, 26, 9, 80, 10, 242, 12, 728, 27, 2186, 28, 6560, 30, 19682, 31, 59048, 36, 177146, 37, 531440, 39, 1594322, 81, 4782968, 82, 14348906, 84, 43046720, 85, 129140162, 90, 387420488, 91, 1162261466, 93

Offset: 1

R. J. Mathar, Mar 31 2009

Numbers of A031944 do not appear in this sequence. After a number which has base-3 digits 0 and 1, a number of the form 3^k-1 (see A024023) follows by definition, because its base-3 digits are all 2.

			The 4th term cannot be 4 because 4(base10)=11(base3) shares a common digit 1 with a(3)=3(base10)=10(base3). It cannot be 5(base10)=12(base3) because this shares the digit 1 with 3=10(base3). It cannot be 6(base10)=20(base3) because this shares the digit 0 with 3=10(base3). It cannot be 7(base10)=21(base3) because this shares the digit 1 with 3=10(base3). It becomes a(4)=8(base10)=22(base3) which does not have the digit 0 or 1 of a(3)=10(base3).

Cf. A067581 (base-10), A158929 (base-4), A158930 (base-5).

cp's OEIS Frontend

A049354 Digitally balanced numbers in base 3: equal numbers of 0's, 1's, 2's.

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

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A212193 In ternary representation of n: a(n) = if n is pandigital then 3 else least digit not used.

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A043530 Number of distinct base-3 digits of n.

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A158928 a(n) is the smallest integer not yet in the sequence with no common base-3 digit with a(n-1).

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