A081607 -id:A081607 - cp's OEIS frontend

A077267 Number of zeros in base-3 expansion of n.

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

Offset: 0

Henry Bottomley, Nov 01 2002

			a(8)=0 since 8 written in base 3 is 22 with 0 zeros;
a(9)=2 since 9 written in base 3 is 100 with 2 zeros;
a(10)=1 since 10 written in base 3 is 101 with 1 zero.

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
Eric Weisstein's World of Mathematics, Ternary.
Wikipedia, Ternary numeral system

Cf. A023416, A077266, A062756, A081603.

Cf. A007089, A081605, A032924, A081607, A081608, A077267, A134023. - Reinhard Zumkeller, Mar 23 2003

a077267 n = a079978 n + if n < 3 then 0 else a077267 (n `div` 3)
-- Reinhard Zumkeller, Feb 21 2013

Table[Count[IntegerDigits[n,3],0],{n,0,6!}] (* Vladimir Joseph Stephan Orlovsky, Jul 25 2009 *)
DigitCount[Range[0,110],3,0] (* Harvey P. Dale, Jul 04 2021 *)

a(1)=a(2)=0; a(3n)=a(n)+1; a(3n+1)=a(3n+2)=a(n). a(3^n-2)=a(3^n-1)=0; a(3^n)=n. a(n)=A077266(n, 3).
a(n) + A062756(n) + A081603(n) = A081604(n). - Reinhard Zumkeller, Mar 23 2003
G.f.: (Sum_{k>=0} x^(3^(k+1))/(1 + x^(3^k) + x^(2*3^k)))/(1-x). - Franklin T. Adams-Watters, Nov 03 2005
a(n) = A079978(n) if n < 3, A079978(n) + a(floor(n/3)) otherwise. - Reinhard Zumkeller, Feb 21 2013

a(0)=1 added, offset changed to 0 and b-file adjusted by Reinhard Zumkeller, Feb 21 2013

Wrong formula deleted by Reinhard Zumkeller, Feb 21 2013

A081605 Numbers having at least one 0 in their ternary representation.

Original entry on oeis.org

0, 3, 6, 9, 10, 11, 12, 15, 18, 19, 20, 21, 24, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 42, 45, 46, 47, 48, 51, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 69, 72, 73, 74, 75, 78, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100

Offset: 1

Reinhard Zumkeller, Mar 23 2003

Complement of A032924.

A212193(a(n)) <> 0. [Reinhard Zumkeller, May 04 2012]

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A007089, A077267, A081607, A081606, A074940.

import Data.List (findIndices)
a081605 n = a081605_list !! (n-1)
a081605_list = findIndices (/= 0) a212193_list
-- Reinhard Zumkeller, May 04 2012

Select[Range[0,100],DigitCount[#,3,0]>0&] (* Harvey P. Dale, Aug 10 2021 *)

from itertools import count, islice
def A081605_gen(): # generator of terms
    a = -1
    for n in count(2):
        b = int(bin(n)[3:],3) + (3**(n.bit_length()-1)-1>>1)
        yield from range(a+1,b)
        a = b
A081605_list = list(islice(A081605_gen(),30)) # Chai Wah Wu, Oct 13 2023

A081610 Number of numbers <= n having at least one 2 in their ternary representation.

Original entry on oeis.org

0, 0, 1, 1, 1, 2, 3, 4, 5, 5, 5, 6, 6, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 19, 19, 20, 20, 20, 21, 22, 23, 24, 24, 24, 25, 25, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53

Offset: 0

Reinhard Zumkeller, Mar 23 2003

a(n) + A081611(n) = n+1. Partial sums of A189820.

Nathaniel Johnston, Table of n, a(n) for n = 0..10000

Cf. A007089, A081603, A081607, A081609, A189820.

num2tern := proc(n) return numboccur(convert(n,base,3),2): end: a:=0: for n from 0 to 80 do a:=a+`if`(num2tern(n)>0,1,0): printf("%d, ",a): od: # Nathaniel Johnston, May 17 2011

Accumulate[Table[If[DigitCount[n,3,2]>0,1,0],{n,0,70}]] (* Harvey P. Dale, Aug 20 2012 *)

first(n)=my(s,t); vector(n,k,t=Set(digits(k,3)); s+=t[#t]==2) \\ Charles R Greathouse IV, Sep 02 2015

from gmpy2 import digits
def A081610(n):
    l = (s:=digits(n,3)).find('2')
    if l >= 0: s = s[:l]+'1'*(len(s)-l)
    return n-int(s,2) # Chai Wah Wu, Dec 05 2024

a(n) ~ n. - 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

A081609 Number of numbers <= n having at least one 1 in their ternary representation.

Original entry on oeis.org

0, 1, 1, 2, 3, 4, 4, 5, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 14, 15, 15, 16, 17, 18, 18, 19, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 46, 47, 47, 48, 49, 50, 50, 51, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 60, 61

Offset: 0

Reinhard Zumkeller, Mar 23 2003

nonn

a(n) + A061392(n) = n+1.

Cf. A007089, A062756, A081607, A081610, A061392.

Partial sums of A316829.

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

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A077267 Number of zeros in base-3 expansion of n.

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A081605 Numbers having at least one 0 in their ternary representation.

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A081610 Number of numbers <= n having at least one 2 in their ternary representation.

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

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A081609 Number of numbers <= n having at least one 1 in their ternary representation.

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