A077267 -id:A077267 - cp's OEIS frontend

A077266 Triangle of number of zeros when n is written in base k (2 <= k <= n).

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

Offset: 2

Henry Bottomley, Nov 01 2002

			Rows start:
  1;
  0,1;
  2,0,1;
  1,0,0,1;
  1,1,0,0,1;
  0,0,0,0,0,1;
  3,0,1,0,0,0,1;
  2,2,0,0,0,0,0,1;
  etc.
9 can be written in bases 2-9 as: 1001, 100, 21, 14, 13, 12, 11 and 10, in which case the numbers of zeros are 2,2,0,0,0,0,0,1.

Harvey P. Dale, Table of n, a(n) for n = 2..1000

Columns include A023416 and A077267. Row sums are A033093, row maxima are A062842, number of positive terms in each row are A077268.

Table[Count[#,0]&/@IntegerDigits[n,Range[2,n]],{n,2,15}]//Flatten (* Harvey P. Dale, Jun 02 2025 *)

T(n, k) = #select(x->(x==0), digits(n, k));
row(n) = vector(n-1, k, T(n,k+1));
tabl(nn) = for (n=2, nn, print(row(n))); \\ Michel Marcus, Sep 02 2020

T(nk, k)=T(n, k)+1; T(nk+m, k)=T(n, k) if 0

A330166 Length of the longest run of 0's in the ternary expression of n.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 1, 0, 0, 2, 1, 1, 1, 0, 0, 1, 0, 0, 2, 1, 1, 1, 0, 0, 1, 0, 0, 3, 2, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 0, 0, 1, 0, 0, 2, 1, 1, 1, 0, 0, 1, 0, 0, 3, 2, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 0, 0, 1, 0, 0, 2, 1, 1, 1, 0, 0, 1, 0, 0, 4, 3, 3, 2, 2, 2, 2, 2, 2, 2

Offset: 0

Joshua Oliver, Dec 04 2019

All numbers appear in this sequence. The n-th power of 3 (A000244(n)) has n 0's in its ternary expression.

The longest run of zeros possible in this sequence is 2, as the last digit of the ternary expression of the integers cycles between 0, 1, and 2, meaning that at least one of three consecutive numbers has a 0 in its ternary expression.

			For n = 87, the ternary expression of 87 is 10020. The length of the runs of 0's in the ternary expression of 87 are 2 and 1, respectively. The larger of these two values is 2, so a(87) = 2.
   n [ternary n] a(n)
   0 [        0] 1
   1 [        1] 0
   2 [        2] 0
   3 [      1 0] 1
   4 [      1 1] 0
   5 [      1 2] 0
   6 [      2 0] 1
   7 [      2 1] 0
   8 [      2 2] 0
   9 [    1 0 0] 2
  10 [    1 0 1] 1
  11 [    1 0 2] 1
  12 [    1 1 0] 1
  13 [    1 1 1] 0
  14 [    1 1 2] 0
  15 [    1 2 0] 1
  16 [    1 2 1] 0
  17 [    1 2 2] 0
  18 [    2 0 0] 2
  19 [    2 0 1] 1
  20 [    2 0 2] 1

Wikipedia, Ternary numeral system.

Cf. A000244, A007089, A077267, A087117, A330036, A330167, A330168.

Equals zero iff n is in A032924.

Table[Max@FoldList[If[#2==0,#1+1,0]&,0,IntegerDigits[n,3]],{n,0,90}]

a(A000244(n)) = a(3^n) = n.

a(n) = 0 iff n is in A032924.

A365278 In the binary expansion of n replace each run of k consecutive 1's by the decimal digits of A007931(k) to get the ternary expansion of a(n).

Original entry on oeis.org

0, 1, 3, 2, 9, 10, 6, 4, 27, 28, 30, 11, 18, 19, 12, 5, 81, 82, 84, 29, 90, 91, 33, 31, 54, 55, 57, 20, 36, 37, 15, 7, 243, 244, 246, 83, 252, 253, 87, 85, 270, 271, 273, 92, 99, 100, 93, 32, 162, 163, 165, 56, 171, 172, 60, 58, 108, 109, 111, 38, 45, 46, 21

Offset: 0

Rémy Sigrist, Aug 30 2023

This sequence is a permutation of the nonnegative integers with inverse A365279.
For any pair (b, c) of bases >= 2, we can devise a similar sequence, say F_{b, c}:
- for any d >= 2, let Z_d be the set of zeroless numbers in base d,
- in the base b expansion of n replace each run of consecutive nonzero digits (say corresponding to Z_b(k) for some k > 0) by the base c digits of Z_c(k) to get the base c expansion of F_{b, c}(n),
- F_{b, c} is a permutation of the nonnegative integers with inverse F_{c, b},
- F_{c, d} o F_{b, c} = F_{b, d} and F_{b, b} is the identity,
- in particular the present sequence corresponds to F_{2, 3} and its inverse to F_{3, 2}.

			The binary expansion of 415 is "110011111", A007931(2) = 2 and A007931(5) = 21, so the ternary expansion of a(415) is "20021", and a(415) = 169.

Rémy Sigrist, Table of n, a(n) for n = 0..8192
Rémy Sigrist, PARI program
Index entries for sequences related to binary expansion of n
Index entries for sequences that are permutations of the natural numbers

Cf. A007931, A023416, A032924, A077267, A290308, A365279 (inverse).

A007931[n_]:=Rest[IntegerDigits[n+1,2]]+1;
A365278[n_]:=FromDigits[Flatten[Map[If[First[#]==1,A007931[Length[#]],#]&,Split[IntegerDigits[n,2]]]],3];
Array[A365278,100,0] (* Paolo Xausa, Oct 17 2023 *)

PARI
```
See Links section.
```

a(2*n) = 3*a(n).
a(2^k - 1) = A032924(k) for any k > 0.
A077267(a(n)) = A023416(n).

A370860 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

1, 3, 4, 10, 12, 13, 14, 16, 22, 28, 30, 31, 32, 34, 36, 37, 38, 39, 40, 41, 42, 43, 46, 48, 49, 58, 64, 66, 67, 85, 86, 88, 91, 92, 93, 94, 95, 96, 97, 100, 102, 103, 109, 110, 111, 112, 113, 114, 115, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127

Offset: 1

Clark Kimberling, Mar 03 2024

			The ternary representation of 16 is 121, for which c(0)=0 <= c(1)=2 > c(2)=1.

Cf. A007089, A077267, A062756, A081603.

Cf. A370853, A370859, A370861, A370862.

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

A370861 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

1, 4, 5, 7, 10, 12, 13, 14, 16, 22, 31, 32, 34, 37, 38, 39, 40, 41, 42, 43, 44, 46, 48, 49, 50, 52, 58, 64, 66, 67, 68, 70, 76, 85, 91, 93, 94, 95, 97, 98, 103, 104, 106, 109, 111, 112, 113, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 127, 128

Offset: 1

Clark Kimberling, Mar 03 2024

			The ternary representation of 16 is 121, for which c(0)=0 < c(1)=2 >= c(2)=1.

Cf. A007089, A077267, A062756, A081603.

Cf. A370853, A370859, A370860, A370862.

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

A370862 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

1, 3, 4, 5, 7, 10, 11, 12, 13, 14, 15, 16, 19, 21, 22, 28, 30, 31, 32, 34, 36, 37, 38, 39, 40, 41, 42, 43, 44, 46, 48, 49, 50, 52, 58, 64, 66, 67, 68, 70, 76, 85, 86, 88, 91, 92, 93, 94, 95, 96, 97, 98, 100, 102, 103, 104, 106, 109, 110, 111, 112, 113, 114

Offset: 1

Clark Kimberling, Mar 09 2024

			The ternary representation of 15 is 120, for which c(0)=1 <= c(1)=1 >= c(2)=1.

Cf. A007089, A077267, A062756, A081603.

Cf. A370853, A370859, A370860, A370861.

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

A370866 Positive integers 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

2, 6, 8, 11, 15, 18, 19, 20, 21, 24, 26, 29, 33, 35, 45, 47, 51, 54, 55, 56, 57, 59, 60, 61, 62, 63, 65, 69, 72, 73, 74, 75, 78, 80, 83, 87, 89, 99, 101, 105, 107, 135, 137, 141, 143, 153, 155, 159, 162, 163, 164, 165, 167, 168, 169, 170, 171, 173, 177, 179

Offset: 1

Clark Kimberling, Mar 11 2024

			The ternary representation of 20 is 202, for which c(0)=1 >= c(1)=0 <= c(2)=2.

Cf. A007089, A077267, A062756, A081603.

Cf. A370853, A370859, A370863, A370864, A370865.

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

A370871 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

3, 9, 27, 28, 30, 36, 81, 82, 84, 86, 88, 90, 92, 96, 100, 102, 108, 110, 114, 126, 136, 138, 144, 166, 172, 174, 190, 192, 198, 243, 244, 246, 247, 248, 250, 252, 253, 254, 255, 258, 262, 264, 270, 271, 272, 273, 276, 279, 288, 298, 300, 306, 324, 325, 326

Offset: 1

Clark Kimberling, Mar 11 2024

			The ternary representation of 84 is 10010, for which c(0)=3 >= c(1)=2 > c(2)=0.

Cf. A007089, A077267, A062756, A081603.

Cf. A370870, A370872, A370873.

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

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A082555 Primes whose base-3 representation does not contain a 0.

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A370864 Numbers m such that c(0) >= c(1) < c(2), where c(k) = number of k's in the ternary representation of m.

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A077266 Triangle of number of zeros when n is written in base k (2 <= k <= n).

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A330166 Length of the longest run of 0's in the ternary expression of n.

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A365278 In the binary expansion of n replace each run of k consecutive 1's by the decimal digits of A007931(k) to get the ternary expansion of a(n).

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A370860 Numbers m such that c(0) <= c(1) > c(2), where c(k) = number of k's in the ternary representation of m.

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A370861 Numbers m such that c(0) < c(1) >= c(2), where c(k) = number of k's in the ternary representation of m.

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A370862 Numbers m such that c(0) <= c(1) >= c(2), where c(k) = number of k's in the ternary representation of m.

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A370866 Positive integers m such that c(0) >= c(1) <= c(2), where c(k) = number of k's in the ternary representation of m.