A077267 -id:A077267 - cp's OEIS frontend

A370859 Numbers m such that c(0) < c(1) > c(2), where c(k) = number of k's in the ternary representation of m.

1, 4, 10, 12, 13, 14, 16, 22, 31, 32, 34, 37, 38, 39, 40, 41, 42, 43, 46, 48, 49, 58, 64, 66, 67, 85, 91, 93, 94, 95, 97, 103, 109, 111, 112, 113, 115, 117, 118, 119, 120, 121, 122, 123, 124, 125, 127, 129, 130, 131, 133, 139, 145, 147, 148, 149, 151, 157

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. A355275, A370853, A370860, A370861, A370862, A370863, A370870.

filter:= proc(n) local L,m;
L:= convert(n,base,3); m:= numboccur(1,L);
numboccur(0,L) < m and numboccur(2,L) < m
end proc:
select(filter, [$1 .. 200]); # Robert Israel, Mar 03 2024

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

A104320 Number of zeros in ternary representation of 2^n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 1, 2, 2, 1, 1, 1, 4, 1, 0, 4, 2, 3, 3, 3, 3, 3, 7, 7, 9, 5, 6, 6, 4, 4, 3, 5, 6, 7, 9, 9, 10, 6, 6, 9, 9, 8, 9, 8, 7, 13, 12, 13, 9, 5, 9, 8, 6, 16, 13, 9, 10, 11, 11, 7, 14, 13, 13, 9, 12, 14, 15, 15, 11, 11, 17, 15, 19, 14, 19, 12, 18, 15, 11, 10, 16, 15, 14, 14, 13, 17, 14

Offset: 0

Reinhard Zumkeller, Mar 01 2005

Conjecture from N. J. A. Sloane: a(n) > 0 for n > 15, see A102483.

			n=13: 2^13=8192 -> '102020102', a(13) = 4.

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A004642, A007089.

[Multiplicity(Intseq(2^n,3),0):n in [0..90]]; // Marius A. Burtea, Nov 17 2019

f:= n -> numboccur(0, convert(2^n,base,3)):
map(f, [$0..100]); # Robert Israel, Nov 17 2019

Table[DigitCount[2^n,3,0],{n,0,90}] (* Harvey P. Dale, May 06 2014 *)

a(n) = my(d=vecsort(digits(2^n, 3))); #setintersect(d, vector(#d)) \\ Felix Fröhlich, Nov 17 2019

a(n) = #select(d->!d, digits(2^n, 3)); \\ Ruud H.G. van Tol, May 09 2024

a(n) = A077267(A000079(n)).

a(A104321(n))=n and a(m)<>n for m < A104321(n).

A188341 Numbers having no 0's and not more than one 1 in their representation in base 3.

Original entry on oeis.org

1, 2, 5, 7, 8, 17, 23, 25, 26, 53, 71, 77, 79, 80, 161, 215, 233, 239, 241, 242, 485, 647, 701, 719, 725, 727, 728, 1457, 1943, 2105, 2159, 2177, 2183, 2185, 2186, 4373, 5831, 6317, 6479, 6533, 6551, 6557, 6559, 6560, 13121, 17495, 18953

Offset: 1

Vladimir Shevelev, Apr 02 2011

Alois P. Heinz, Table of n, a(n) for n = 1..1000
V. Shevelev, Binomial Coefficient Predictors, Journal of Integer Sequences, Vol. 14 (2011), Article 11.2.8

Cf. A062756, A077267.

a:= proc(n) option remember; local i, l, m, t;
      m:= `if`(n=1, 0, a(n-1));
      l:= NULL;
      for t while m>0 do l:= l, irem(m, 3, 'm') od;
      l:= array([l, 0]);
      for i while l[i]=2 do od;
      if l[i]=0 then l[i]:= 1
                else l[i]:= 2;
                     if i>1 then l[i-1]:= 1 fi
      fi;
      add(l[i] *3^(i-1), i=1..t)
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Apr 02 2011

okQ[n_]:=DigitCount[n,3,0]==0&&DigitCount[n,3,1]<2; Select[Range[20000], okQ]  (* Harvey P. Dale, Apr 16 2011 *)

A081607 Number of numbers <= n having at least one 0 in their ternary representation.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 5, 6, 7, 7, 7, 8, 8, 8, 9, 10, 11, 12, 12, 12, 13, 13, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 26, 26, 27, 27, 27, 28, 29, 30, 31, 31, 31, 32, 32, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 45, 45, 46, 46, 46, 47, 48, 49, 50

Offset: 0

Reinhard Zumkeller, Mar 23 2003

nonn

a(n) + A081608(n) = n+1.

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A007089, A077267, A081609, A081610.

f:= n -> `if`(has(convert(n,base,3),0),1,0):
ListTools:-PartialSums(map(f, [$0..100])); # Robert Israel, Mar 18 2018

Accumulate[Boole[Table[DigitCount[n,3,0]>0,{n,0,80}]]] (* Harvey P. Dale, Jun 23 2017 *)

first(n)=my(s,t); vector(n,k, t=Set(digits(k,3)); s+=(t[1]==0)) \\ Charles R Greathouse IV, Sep 02 2015

A081608 Number of numbers <= n having no 0 in their ternary representation.

Original entry on oeis.org

0, 1, 2, 2, 3, 4, 4, 5, 6, 6, 6, 6, 6, 7, 8, 8, 9, 10, 10, 10, 10, 10, 11, 12, 12, 13, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 15, 16, 16, 17, 18, 18, 18, 18, 18, 19, 20, 20, 21, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 23, 24, 24, 25, 26, 26, 26, 26

Offset: 0

Reinhard Zumkeller, Mar 23 2003

nonn

a(n) + A081607(n) = n+1.

Winston de Greef, Table of n, a(n) for n = 0..16383

Cf. A007089, A077267, A061392, A081611.

Accumulate[Table[If[DigitCount[n,3,0]==0,1,0],{n,0,80}]] (* Harvey P. Dale, Oct 21 2024 *)

first(n)=my(s,t); vector(n,k, t=Set(digits(k,3)); s+=!!t[1]) \\ Charles R Greathouse IV, Sep 02 2015

A370863 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

6, 18, 20, 24, 54, 56, 60, 62, 72, 74, 78, 89, 101, 105, 137, 141, 153, 162, 164, 167, 168, 169, 170, 173, 177, 180, 181, 182, 183, 186, 188, 191, 195, 207, 216, 217, 218, 219, 222, 224, 225, 234, 236, 240, 251, 263, 267, 269, 299, 303, 305, 315, 317, 321

Offset: 1

Clark Kimberling, Mar 09 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, A370864, A370865, A370866, A370870.

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

A370870 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

9, 27, 81, 82, 84, 90, 108, 243, 244, 246, 248, 250, 252, 254, 258, 262, 264, 270, 272, 276, 288, 298, 300, 306, 324, 326, 330, 342, 378, 406, 408, 414, 432, 490, 496, 498, 514, 516, 522, 568, 570, 576, 594, 729, 730, 732, 733, 734, 736, 738, 739, 740, 741

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. A370853, A370859, A370863, A370864, A370871, A370872, A370873.

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

A073779 Number of 0's in base-3 representation of n-th prime.

Original entry on oeis.org

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

Offset: 1

Zak Seidov, Aug 11 2002

a(n) = 0 if prime(n) is in A082555. - Robert Israel, Dec 28 2018

			a(10)=2 as 10th prime is 29 in base-10 representation, or 1002 in base-3 representation.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A073780, A073781, A077267, A082555.

[ Multiplicity({* a: a in Intseq(p, 3) *}, 0): p in PrimesUpTo(600) ]; // Klaus Brockhaus, Oct 10 2010

f:= n -> numboccur(0,convert(ithprime(n),base,3)):
map(f, [$1..200]); # Robert Israel, Dec 28 2018

A073779[n_] := Length[Cases[IntegerDigits[Prime[n], 3], 0]];
DigitCount[#,3,0]&/@Prime[Range[120]]  (* Harvey P. Dale, Apr 26 2011 *)

a(n) = A077267(A000040(n)). - Michel Marcus, Oct 02 2013

More terms from Klaus Brockhaus, Oct 10 2010

A160380 a(0) = 0; for n >= 1, a(n) = number of 0's in base-4 representation of n.

Original entry on oeis.org

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

Offset: 0

Frank Ruskey, Jun 05 2009

The base-4 representation of 0 is 0, and contains a single zero. - N. J. A. Sloane, Apr 26 2021

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
F. T. Adams-Watters, F. Ruskey, Generating Functions for the Digital Sum and Other Digit Counting Sequences, JIS 12 (2009) 09.5.6

Cf. A007090, A023416, A077267.

import Data.List (unfoldr)
a160380 = sum . map ((0 ^ ) . (`mod` 4)) .
   unfoldr (\x -> if x == 0 then Nothing else Just (x, x `div` 4))
-- Reinhard Zumkeller, Apr 22 2011

Join[{0},Table[DigitCount[n,4,0],{n,110}]] (* Harvey P. Dale, Oct 18 2015 *)

a(n) = #select(x->(x==0), digits(n, 4)); \\ Michel Marcus, Apr 26 2021

Recurrence relation: a(0) = 0, a(4m) = 1+a(m), a(4m+1) = a(4m+2) = a(4m+3) = a(m).

Generating function: (1/(1-z))*Sum_{m>=0} (z^(4^(m+1))*(1 - z^(4^m))/(1 - z^(4^(m+1)))).

Definition clarified by Georg Fischer, Apr 26 2021

A077268 Number of bases in which n requires at least one zero to be written.

Original entry on oeis.org

0, 1, 1, 2, 2, 3, 1, 3, 3, 4, 3, 5, 2, 3, 3, 4, 3, 6, 4, 6, 4, 3, 2, 7, 3, 4, 5, 7, 4, 7, 2, 6, 5, 5, 6, 8, 4, 5, 5, 8, 3, 7, 2, 5, 6, 4, 3, 9, 4, 7, 7, 7, 4, 9, 6, 8, 4, 4, 3, 11, 3, 4, 5, 7, 7, 9, 4, 6, 6, 9, 4, 11, 5, 6, 8, 7, 7, 9, 4, 9, 6, 6, 5, 12, 6, 5, 5, 9, 4, 11, 5, 6, 4, 4, 5, 11, 4, 7, 8, 10, 6

Offset: 1

Henry Bottomley, Nov 01 2002

			a(9)=3 since it requires zeros when written in bases 2, 3 or 9 (as 1001, 100 or 10 respectively).

Eric M. Schmidt, Table of n, a(n) for n = 1..10000

Cf. A033093, A062842, A077266, A077267.

a(n) = sum(i=2, n, ! vecmin(digits(n, i))); \\ Michel Marcus, Jul 09 2014

def A077268(n) : return sum(0 in n.digits(m) for m in range(2,n+1)) # Eric M. Schmidt, Jul 09 2014

cp's OEIS Frontend

A370859 Numbers m such that c(0) < c(1) > c(2), where c(k) = number of k's in the ternary representation of m.

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Mathematica

A104320 Number of zeros in ternary representation of 2^n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

PARI

Formula

A188341 Numbers having no 0's and not more than one 1 in their representation in base 3.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

A081607 Number of numbers <= n having at least one 0 in their ternary representation.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A081608 Number of numbers <= n having no 0 in their ternary representation.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A370863 Numbers m such that c(0) > c(1) < c(2), where c(k) = number of k's in the ternary representation of m.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

A370870 Numbers m such that c(0) > c(1) > c(2), where c(k) = number of k's in the ternary representation of m.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

A073779 Number of 0's in base-3 representation of n-th prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs