A032924 -id:A032924 - cp's OEIS frontend

A363428 Squares whose base-3 expansion has no 0.

1, 4, 16, 25, 49, 121, 400, 484, 1444, 1849, 3364, 5476, 10201, 10609, 10816, 12769, 17161, 19321, 19600, 155236, 169744, 274576, 286225, 344569, 346921, 450241, 502681, 863041, 885481, 984064, 1042441, 4008004, 4190209, 4596736, 7203856, 7263025, 7706176, 12752041, 14326225, 14341369, 23833924

Offset: 1

Robert Israel, Jun 01 2023

			a(5) = 49 is a term because 49 = 7^2 = 1121_3.

Chai Wah Wu, Table of n, a(n) for n = 1..3370 (terms 1..254 from Robert Israel)

Intersection of A000290 and A032924. Cf. A363408.

R1:= {1};
S1:= {1,2};
for i from 1 to 15 do
  S1:= map(t -> (3*t+1, 3*t+2), S1);
  R1:= R1 union select(issqr,S1);
od:
Rl;

Select[Range[5000]^2, ! MemberQ[IntegerDigits[#, 3], 0] &] (* Amiram Eldar, Jun 01 2023 *)

from itertools import islice, count
from gmpy2 import digits
def A363428_gen(): # generator of terms
    return filter(lambda n:'0' not in digits(n,3), (n**2 for n in count(0)))
A363428_list = list(islice(A363428_gen(),20)) # Chai Wah Wu, Jun 17 2023

A365279 Inverse permutation to A365278.

Original entry on oeis.org

0, 1, 3, 2, 7, 15, 6, 31, 63, 4, 5, 11, 14, 127, 255, 30, 511, 1023, 12, 13, 27, 62, 2047, 4095, 126, 8191, 16383, 8, 9, 19, 10, 23, 47, 22, 95, 191, 28, 29, 59, 254, 32767, 65535, 510, 131071, 262143, 60, 61, 123, 1022, 524287, 1048575, 2046, 2097151, 4194303

Offset: 0

Rémy Sigrist, Aug 31 2023

			A365278(42) = 273, hence a(273) = 42.

Rémy Sigrist, Table of n, a(n) for n = 0..6561
Rémy Sigrist, PARI program
Index entries for sequences that are permutations of the natural numbers

Cf. A023416, A032924, A077267, A365278.

PARI
```
See Links section.
```

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

A382412 Numbers with no zeros in their base-7 representation.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27, 29, 30, 31, 32, 33, 34, 36, 37, 38, 39, 40, 41, 43, 44, 45, 46, 47, 48, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 68, 69, 71, 72, 73, 74, 75, 76, 78, 79, 80, 81, 82, 83, 85, 86, 87, 88, 89, 90

Offset: 1

Paolo Xausa, Mar 24 2025

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. zeroless numbers in other bases: A126646 (base 2), A032924 (base 3), A023705 (base 4), A023721 (base 5), A248910 (base 6), A255805 (base 8), A255808 (base 9), A052382 (base 10).

Cf. A007093, A043393, A249102, A382413 (complement).

Select[Range[100], DigitCount[#, 7, 0] == 0 &]

A259566 Numbers following gaps in the sequence of base-3 numbers that don't contain 0.

Original entry on oeis.org

1, 4, 7, 13, 16, 22, 25, 40, 43, 49, 52, 67, 70, 76, 79, 121, 124, 130, 133, 148, 151, 157, 160, 202, 205, 211, 214, 229, 232, 238, 241, 364, 367, 373, 376, 391, 394, 400, 403, 445, 448, 454, 457, 472, 475, 481, 484, 607, 610, 616, 619, 634, 637, 643, 646, 688, 691, 697, 700, 715, 718, 724, 727, 1093, 1096, 1102, 1105, 1120

Offset: 1

Sean Oneil, Jun 30 2015

Partial sums for the convergent modified harmonic series in base 3 excluding 0 = Sum of 1/a(n) + 1/(a(n) + 1) = Sum of (2*a(n) + 1)/(a(n)*(a(n) + 1)).

			Pattern of numbers of skipped terms (numbers in base 3 with at least one zero) is 1 (3 = 10_3), 1 (6 = 20_3), 3+1 (9 = 100_3, 10 = 101_3, 11 = 102_3, 12 = 110_3), 1, 3+1, 1, 9+3+1, 1, 3+1, 1, 9+3+1, 1, 3+1, 1, 27+9+3+1, ...

Robert Baillie, Sums of Reciprocals of Integers Missing a Given Digit, American Mathematical Monthly (Washington, DC: Mathematical Association of America) 86 (5): 372-374, May 1979, doi:10.2307/2321096. ISSN 0002-9890. JSTOR 2321096.

Cf. A032924.
Subset of A016777 (congruent to 1 mod 3).
Each term is one more than each number that follows a gap in A081605.

lista(nn)=prec0 = 1; for(n=1, nn, if (vecmin(digits(n, 3)), if (prec0, print1(n,, ", ")); prec0 = 0, prec0 = 1);); \\ Michel Marcus, Aug 03 2015

def A259566(n): return int(bin(m:=n)[3:],3)*3 + (3**m.bit_length()-1>>1) if n>1 else 1 # Chai Wah Wu, Oct 13 2023

a(n) = A032924(2n - 1).

A266001 Numbers with no 0's in their base 3 and base 4 expansions.

Original entry on oeis.org

1, 2, 5, 7, 13, 14, 22, 23, 25, 26, 41, 43, 53, 121, 122, 125, 149, 151, 157, 158, 214, 215, 229, 230, 233, 238, 239, 365, 367, 373, 374, 377, 445, 446, 473, 475, 485, 607, 617, 619, 634, 635, 637, 638, 697, 698, 701, 725, 727, 1366, 1367, 1373, 1375, 1429, 1430, 1445, 1447, 1453, 1454

Offset: 1

Robin Powell, Jan 27 2016

Intersection of A023705 and A032924.

1, 7 and 32767 also share this property in base 2.

			53 is 1222 in base 3 and 311 in base 4; no zeros are shown in either representation and so 53 is a term.
Similarly, 121 is 11111 in base 3 and 1321 in base 4 so it is also a term.

Chai Wah Wu, Table of n, a(n) for n = 1..12682

Cf. A023705, A032924, A262958, A262963.

isokd(n) = vecmin(digits(n, 3)) && vecmin(digits(n, 4)); \\ Michel Marcus, Jan 28 2016

from _future_ import division
from gmpy2 import digits
A266001_list = [j for j in (int(format(i,'b'),3)+(3**n-1)//2 for n in range(1,10) for i in range(2**n)) if '0' not in digits(j,4)] # Chai Wah Wu, Feb 13 2016

A337296 Number whose sum and product of ternary representation digits are equal.

Original entry on oeis.org

0, 1, 2, 8, 134, 152, 158, 160, 206, 212, 214, 230, 232, 238, 265760, 265814, 265832, 265838, 265840, 265976, 265994, 266000, 266002, 266048, 266054, 266056, 266072, 266074, 266080, 266462, 266480, 266486, 266488, 266534, 266540, 266542, 266558, 266560, 266566

Offset: 1

Amiram Eldar, Aug 21 2020

In ternary representation all the terms except 0 are zeroless (A032924).
If k is the number of digits 2 of a term, then the number of digits 1 is 2^k - 2*k, and the total number of digits is thus 2^k - k (A000325).
The total number of terms with k digits 2, for k = 1, 2, ..., is binomial(2^k-k,k) = 1, 1, 10, 495, 80730, 40475358, ...

			8 is a term since in base 3 the representation of 8 is 22 and 2 + 2 = 2 * 2.

Amiram Eldar, Table of n, a(n) for n = 1..10000

The ternary version of A034710.

Cf. A000325, A032924.

Select[Range[0, 266566], Times @@ (d = IntegerDigits[#, 3]) == Plus @@ d &]
(* or *)
f[k_] := FromDigits[#, 3] & /@ Permutations[Join[Table[1, {2^k - 2*k}], Table[2, k]]]; Flatten@ Join[{0}, Table[f[k], {k, 0, 4}]] (* Amiram Eldar, Oct 16 2023 *)

isok(m) = my(d=digits(m,3)); vecsum(d) == vecprod(d); \\ Michel Marcus, Aug 22 2020

A348706 Delete all 0's from ternary expansion of n.

Original entry on oeis.org

1, 2, 1, 4, 5, 2, 7, 8, 1, 4, 5, 4, 13, 14, 5, 16, 17, 2, 7, 8, 7, 22, 23, 8, 25, 26, 1, 4, 5, 4, 13, 14, 5, 16, 17, 4, 13, 14, 13, 40, 41, 14, 43, 44, 5, 16, 17, 16, 49, 50, 17, 52, 53, 2, 7, 8, 7, 22, 23, 8, 25, 26, 7, 22, 23, 22, 67, 68, 23, 70, 71, 8, 25

Offset: 1

Rémy Sigrist, Oct 30 2021

			The first terms, in decimal and in base 3, are:
  n   a(n)  ter(n)  ter(a(n))
  --  ----  ------  ---------
   1     1       1          1
   2     2       2          2
   3     1      10          1
   4     4      11         11
   5     5      12         12
   6     2      20          2
   7     7      21         21
   8     8      22         22
   9     1     100          1
  10     4     101         11
  11     5     102         12
  12     4     110         11
  13    13     111        111
  14    14     112        112
  15     5     120         12

Rémy Sigrist, Table of n, a(n) for n = 1..6561

Cf. A004719 (decimal analog), A032924 (fixed points), A038573 (binary analog).

a[n_] := FromDigits[DeleteCases[IntegerDigits[n, 3], 0], 3]; Array[a, 100] (* Amiram Eldar, Oct 31 2021 *)

a(n, base=3) = fromdigits(select(sign, digits(n, base)), base)

from gmpy2 import digits
def A348706(n): return int(digits(n,3).replace('0',''),3) # Chai Wah Wu, Nov 02 2021

a(n) <= n with equality iff n belongs to A032924.

A363995 Rectangular array by descending antidiagonals: row n consists of the numbers k such that n = 1 + maximal runlength of 0's in the ternary representation of k.

Original entry on oeis.org

1, 2, 3, 4, 6, 9, 5, 10, 18, 27, 7, 11, 28, 54, 81, 8, 12, 29, 82, 162, 243, 13, 15, 36, 83, 244, 486, 729, 14, 19, 45, 108, 245, 730, 1458, 2187, 16, 20, 55, 135, 324, 731, 2188, 4374, 6561, 17, 21, 56, 163, 405, 972, 2189, 6562, 13122, 19683, 22, 24, 63

Offset: 1

Clark Kimberling, Jul 01 2023

Every positive integer occurs exactly once.

			Corner:
    1     2      4     5     7      8     13     14     16     17
    3     6     10    11    12     15     19     20     21     24
    9    18     28    29    36     45     55     56     63     72
   27    54     82    83   108    135    163    164    189    216
   81   162    244   245   324    405    487    488    567    648
  243   486    730   731   972   1215   1459   1460   1701   1944
Let r(n) = maximal runlength of 0s in the ternary representation of n, for n >=1, so that (r(n)) = (0,0,1,0,0,1,0,0,2,...). Thus, r(9)=2, so that the first term in row 3 of the array is 9.

Cf. A000244 (column 1), A032924 (row 1), A363996.

d[n_] := d[n] = First[RealDigits[n, 3]]; f[w_] := FromDigits[w, 3];
s = Map[Split, Table[d[n], {n, 1, 2187}]];
x[n_] := Select[s, MemberQ[#, Table[0, n]] &];
u[n_] := Map[Flatten, x[n]];
t0 = Flatten[Table[FromDigits[#, 3] & /@ Tuples[{1, 2}, n], {n, 5}]];
t = Join[{t0}, Table[Map[f, u[n]], {n, 1, 7}]] ;
TableForm[t]  (* this sequence as an array *)
Table[t[[n - k + 1, k]], {n, 8}, {k, n, 1, -1}] // Flatten (* this sequence *)
(* Next, another program *)
nwCornerD[lists_] := Quiet[Flatten[Reap[NestWhile[# + 1 &, 1, ! {} ===
Sow[Check[lists[[# - Binomial[Floor[1/2 + Sqrt[2*#]], 2]]][[1 - # +
Binomial[Floor[3/2 + Sqrt[2*#]], 2]]], {}]] &[#] &]][[2]]]];
z = 10; radix = 3;
tmp = Map[Max[Map[Count[#, 0] &, #]] &,
   Map[Split, IntegerDigits[Range[radix^z], radix]]];
nwCornerD[Map[Flatten[Position[tmp, #]] &, Range[0, z]]]
(* Peter J. C. Moses, Aug 01 2023 *)

A363996 Rectangular array by descending antidiagonals: row n consists of the numbers k such that n = 1 + maximal runlength of 1's in the ternary representation of k.

Original entry on oeis.org

2, 6, 1, 8, 3, 4, 18, 5, 12, 13, 20, 7, 14, 39, 40, 24, 9, 22, 41, 120, 121, 26, 10, 31, 67, 122, 363, 364, 54, 11, 36, 94, 202, 365, 1092, 1093, 56, 15, 37, 117, 283, 607, 1094, 3279, 3280, 60, 16, 38, 118, 360, 850, 1822, 3281, 9840, 9841, 62, 17, 42, 119

Offset: 1

Clark Kimberling, Jul 01 2023

Every positive integer occurs exactly once.

			Corner:
    2     6     8    18    20    24     26
    1     3     5     7     9    10     11
    4    12    14    22    31    36     37
   13    39    41    67    94   117    118
   40   120   122   202   283   360    361
  121   363   365   607   850   1089  1090
Let r(n) = maximal runlength of 1's in the ternary representation of n, for n >= 1, so that (r(n)) = (1,0,1,2,1,0,1,0,1,...). Thus, r(4)=2, so the first term in row 3 of the array is 4.

Cf. A000244 (column 1), A032924 (row 1), A363995.

d[n_] := d[n] = First[RealDigits[n, 3]]; f[w_] := FromDigits[w, 3];
s = Map[Split, Table[d[n], {n, 1, 50000}]];
x[n_] := Select[s, MemberQ[#, Table[1, n]] &];
u[n_] := Map[Flatten, x[n]];
t0 = Select[Range[1, 4000], DigitCount[#, 3, 1] == 0 &, 20];
v = Table[Take[Map[f, u[n]], Min[{20, Length[u[n]]}]], {n, 1, 11}]
t = Join[{t0}, v]
TableForm[t]  (* this sequence as an array *)
Table[t[[n - k + 1, k]], {n, 11}, {k, n, 1, -1}] // Flatten (* this sequence *)

A371222 Product of digits of (n written in base 3) mod 3.

Original entry on oeis.org

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

Offset: 0

Ctibor O. Zizka, Mar 18 2024

a(A032924(n)) = 1 or 2. For n >= 1, a(A032924(n)) - 1 = A309953(A032924(n)) mod 3 - 1 = A010059(n+1).

			n = 5: 5_10 = 12_3 thus a(5) = 1*2 mod 3 = 2.
n = 8: 8_10 = 22_3 thus a(8) = 2*2 mod 3 = 1.

Cf. A007089, A010059, A032924, A081605, A309953, A371281 (base 10).

a[n_] := Mod[Times @@ IntegerDigits[n, 3], 3]; Array[a, 100, 0] (* Amiram Eldar, Mar 18 2024 *)

from functools import reduce
from sympy.ntheory import digits
def A371222(n): return reduce(lambda a,b: a*b%3,digits(n,3)[1:],1) # Chai Wah Wu, Mar 19 2024

a(n) = A309953(n) mod 3.

a(A081605(n)) = 0.

cp's OEIS Frontend

A363428 Squares whose base-3 expansion has no 0.

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A365279 Inverse permutation to A365278.

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PARI

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A382412 Numbers with no zeros in their base-7 representation.

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Mathematica

A259566 Numbers following gaps in the sequence of base-3 numbers that don't contain 0.

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PARI

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A266001 Numbers with no 0's in their base 3 and base 4 expansions.

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PARI

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A337296 Number whose sum and product of ternary representation digits are equal.

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Mathematica

PARI

A348706 Delete all 0's from ternary expansion of n.

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Mathematica

PARI

Python

Formula

A363995 Rectangular array by descending antidiagonals: row n consists of the numbers k such that n = 1 + maximal runlength of 0's in the ternary representation of k.

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