A062153 -id:A062153 - cp's OEIS frontend

A081604 Number of digits in ternary representation of n.

1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5

Offset: 0

Reinhard Zumkeller, Mar 23 2003

a(n) is the length of row n in table A054635. - Reinhard Zumkeller, Sep 05 2014

			a(8) = 2 because 8 = 22_3, having 2 digits.
a(9) = 3 because 9 = 100_3, having 3 digits.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Ternary.

Cf. A003137, A007089, A030341, A049803, A054635, A062153, A070939, A080342.
Cf. A062756, A077267, A081603.
Cf. A134021, A134025, A134026, A246435.

a081604 n = if n < 3 then 1 else a081604 (div n 3) + 1
-- Reinhard Zumkeller, Sep 05 2014, Feb 21 2013

A081604 := proc(n)
    max(1,1+ilog[3](n)) ;
end proc: # R. J. Mathar, Jul 12 2016

Table[Length[IntegerDigits[n, 3]], {n, 0, 99}] (* Alonso del Arte, Dec 30 2012 *)
Join[{1},IntegerLength[Range[120],3]] (* Harvey P. Dale, Apr 07 2019 *)

a(n) = A062153(n) + 1 for n >= 1.
a(n) = A077267(n) + A062756(n) + A081603(n);
From Reinhard Zumkeller, Oct 19 2007: (Start)
0 <= A134021(n) - a(n) <= 1;
a(A134025(n)) = A134021(A134025(n));
a(A134026(n)) = A134021(A134026(n)) - 1. (End)
a(n+1) = -Sum_{k=1..n} mu(3*k)*floor(n/k). - Benoit Cloitre, Oct 21 2009
a(n) = floor(log_3(n)) + 1. - Can Atilgan and Murat Erşen Berberler, Dec 05 2012
a(n) = if n < 3 then 1 else a(floor(n/3)) + 1. - Reinhard Zumkeller, Sep 05 2014
G.f.: 1 + (1/(1 - x))*Sum_{k>=0} x^(3^k). - Ilya Gutkovskiy, Jan 08 2017

A117967 Positive part of inverse of A117966; write n in balanced ternary and then replace (-1)'s with 2's.

Original entry on oeis.org

0, 1, 5, 3, 4, 17, 15, 16, 11, 9, 10, 14, 12, 13, 53, 51, 52, 47, 45, 46, 50, 48, 49, 35, 33, 34, 29, 27, 28, 32, 30, 31, 44, 42, 43, 38, 36, 37, 41, 39, 40, 161, 159, 160, 155, 153, 154, 158, 156, 157, 143, 141, 142, 137, 135, 136, 140, 138, 139, 152, 150, 151, 146

Offset: 0

Franklin T. Adams-Watters, Apr 05 2006

			7 in balanced ternary is 1(-1)1, changing to 121 ternary is 16, so a(7)=16.

D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 2, pp. 173-175

Alois P. Heinz, Table of n, a(n) for n = 0..9841
Ken Levasseur, The Balanced Ternary Number System

Cf. A117966. a(n) = A004488(A117968(n)). Bisection of A140263. A140267 gives the same sequence in ternary.

a:= proc(n) local d, i, m, r; m:=n; r:=0;
      for i from 0 while m>0 do
         d:= irem(m, 3, 'm');
         if d=2 then m:=m+1 fi;
         r:= r+d*3^i
      od; r
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, May 11 2015

a[n_] := Module[{d, i, m = n, r = 0}, For[i = 0, m > 0, i++, {m, d} = QuotientRemainder[m, 3]; If[d == 2, m++]; r = r + d*3^i]; r];
a /@ Range[0, 100] (* Jean-François Alcover, Jan 05 2021, after Alois P. Heinz *)

from sympy.ntheory.factor_ import digits
def a004488(n): return int("".join([str((3 - i)%3) for i in digits(n, 3)[1:]]), 3)
def a117968(n):
    if n==1: return 2
    if n%3==0: return 3*a117968(n/3)
    elif n%3==1: return 3*a117968((n - 1)/3) + 2
    else: return 3*a117968((n + 1)/3) + 1
def a(n): return 0 if n==0 else a004488(a117968(n)) # Indranil Ghosh, Jun 06 2017

;; Two alternative definitions in MIT/GNU Scheme, defined for whole Z:
(define (A117967 z) (cond ((zero? z) 0) ((negative? z) (A004488 (A117967 (- z)))) (else (let* ((lp3 (expt 3 (A062153 z))) (np3 (* 3 lp3))) (if (< (* 2 z) np3) (+ lp3 (A117967 (- z lp3))) (+ np3 (A117967 (- z np3))))))))
(define (A117967v2 z) (cond ((zero? z) 0) ((negative? z) (A004488 (A117967v2 (- z)))) ((zero? (modulo z 3)) (* 3 (A117967v2 (/ z 3)))) ((= 1 (modulo z 3)) (+ (* 3 (A117967v2 (/ (- z 1) 3))) 1)) (else (+ (* 3 (A117967v2 (/ (+ z 1) 3))) 2))))
;; Antti Karttunen, May 19 2008

a(0) = 0, a(3n) = 3a(n), a(3n+1) = 3a(n)+1, a(3n-1) = 3a(n)+2.

If one adds this clause, then the function is defined on the whole Z: If n<0, then a(n) = A004488(a(-n)) (or equivalently: a(n) = A117968(-n)) and then it holds that a(A117966(n)) = n. - Antti Karttunen, May 19 2008

A102572 a(n) = floor(log_4(n)).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Dec 23 2006

nonn

Cf. A000523, A062153.

Magma
```
[ Ilog(4,n) : n in [1..150] ];
```

a(n)=#digits(n,4)-1 \\ Twice as fast as a(n)=for(i=0,n,(n>>=2)||return(i)); the naïve code a(n)=log(n)\log(4) works for standard realprecision=28 only up to n=4^47-5 and it is slower by another factor 2. - M. F. Hasler, Mar 11 2015

A102572(n)=logint(n,4) \\ M. F. Hasler, Nov 07 2019

G.f.: (1/(1 - x))*Sum_{k>=1} x^(4^k). - Ilya Gutkovskiy, Jan 08 2017

A212445 a(n) = floor( n + log(n) ).

Original entry on oeis.org

1, 2, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 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, 54, 55, 56, 57, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70

Offset: 1

Mohammad K. Azarian, May 17 2012

Complement of A045650. - Michel Marcus, Jun 30 2015

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000523, A062153, A102572, A212446, A212447, A212448, A212449, A212450, A212451, A212452, A212453, A212454.

[Floor(n+Log(n)): n in [1..80]]; // Vincenzo Librandi, Feb 15 2013

Table[Floor[n + Log[n]], {n, 100}] (* T. D. Noe, May 21 2012 *)

a(n)=n+log(n)\1 \\ Charles R Greathouse IV, Aug 07 2012

A081134 Distance to nearest power of 3.

Original entry on oeis.org

0, 1, 0, 1, 2, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7

Offset: 1

Klaus Brockhaus, Mar 08 2003

			a(7) = 2 since 9 is closest power of 3 to 7 and 9 - 7 = 2.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Klaus Brockhaus, Illustration for A081134, A081249, A081250 and A081251
Hsien-Kuei Hwang, S. Janson, and T.-H. Tsai, Exact and asymptotic solutions of the recurrence f(n) = f(floor(n/2)) + f(ceiling(n/2)) + g(n): theory and applications, Preprint 2016.
Hsien-Kuei Hwang, S. Janson, and T.-H. Tsai, Exact and Asymptotic Solutions of a Divide-and-Conquer Recurrence Dividing at Half: Theory and Applications, ACM Transactions on Algorithms, 13:4 (2017), #47; DOI: 10.1145/3127585.
Index entries for sequences related to distance to nearest element of some set

Cf. A002452, A003605, A006166, A053646, A062153, A081249, A081250, A081251.

a:= n-> (h-> min(n-h, 3*h-n))(3^ilog[3](n)):
seq(a(n), n=1..100);  # Alois P. Heinz, Mar 28 2021

Flatten[Table[Join[Range[0,3^n],Range[3^n-1,1,-1]],{n,0,4}]] (* Harvey P. Dale, Dec 31 2013 *)

a(n) = my (p=#digits(n,3)); return (min(n-3^(p-1), 3^p-n)) \\ Rémy Sigrist, Mar 24 2018

def A081134(n):
    kmin, kmax = 0,1
    while 3**kmax <= n:
        kmax *= 2
    while True:
        kmid = (kmax+kmin)//2
        if 3**kmid > n:
            kmax = kmid
        else:
            kmin = kmid
        if kmax-kmin <= 1:
            break
    return min(n-3**kmin, 3*3**kmin-n) # Chai Wah Wu, Mar 31 2021

a(n) = min(n-3^floor(log(n)/log(3)), 3*3^floor(log(n)/log(3))-n).
From Peter Bala, Sep 30 2022: (Start)
a(n) = n - A006166(n); a(n) = 2*n - A003605(n).
a(1) = 0, a(2) = 1, a(3) = 0; thereafter, a(3*n) = 3*a(n), a(3*n+1) = 2*a(n) + a(n+1) and a(3*n+2) = a(n) + 2*a(n+1). (End)

A212454 Ceiling(5n + log(5n)).

Original entry on oeis.org

7, 13, 18, 23, 29, 34, 39, 44, 49, 54, 60, 65, 70, 75, 80, 85, 90, 95, 100, 105, 110, 115, 120, 125, 130, 135, 140, 145, 150, 156, 161, 166, 171, 176, 181, 186, 191, 196, 201, 206, 211, 216, 221, 226, 231, 236, 241, 246, 251, 256, 261, 266, 271, 276, 281

Offset: 1

Mohammad K. Azarian, May 17 2012

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000523, A062153, A102572, A212445-A212453.

[Ceiling(5*n + Log(5*n)): n in [1..80]]; // Vincenzo Librandi, Feb 14 2013

Table[Ceiling[5*n + Log[5*n]], {n, 100}] (* T. D. Noe, May 21 2012 *)

A212453 a(n) = ceiling(4n + log(4n)).

Original entry on oeis.org

6, 11, 15, 19, 23, 28, 32, 36, 40, 44, 48, 52, 56, 61, 65, 69, 73, 77, 81, 85, 89, 93, 97, 101, 105, 109, 113, 117, 121, 125, 129, 133, 137, 141, 145, 149, 153, 158, 162, 166, 170, 174, 178, 182, 186, 190, 194, 198, 202, 206, 210, 214, 218, 222, 226, 230

Offset: 1

Mohammad K. Azarian, May 17 2012

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000523, A062153, A102572, A212445-A212454.

[Ceiling(4*n + Log(4*n)): n in [1..80]]; // Vincenzo Librandi, Feb 14 2013

Table[Ceiling[4*n + Log[4*n]], {n, 100}] (* T. D. Noe, May 21 2012 *)

A212451 Ceiling(2n + log(2n)).

Original entry on oeis.org

3, 6, 8, 11, 13, 15, 17, 19, 21, 23, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125, 127, 129

Offset: 1

Mohammad K. Azarian, May 17 2012

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000523, A062153, A102572, A212445-A212454.

[Ceiling(2*n + Log(2*n)): n in [1..80]]; // Vincenzo Librandi, Feb 14 2013

Table[Ceiling[2*n + Log[2*n]], {n, 100}] (* T. D. Noe, May 21 2012 *)

A212452 Ceiling(3n + log(3n)).

Original entry on oeis.org

5, 8, 12, 15, 18, 21, 25, 28, 31, 34, 37, 40, 43, 46, 49, 52, 55, 58, 62, 65, 68, 71, 74, 77, 80, 83, 86, 89, 92, 95, 98, 101, 104, 107, 110, 113, 116, 119, 122, 125, 128, 131, 134, 137, 140, 143, 146, 149, 152, 156, 159, 162, 165, 168, 171, 174, 177, 180

Offset: 1

Mohammad K. Azarian, May 17 2012

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000523, A062153, A102572, A212445-A212454.

[Ceiling(3*n + Log(3*n)): n in [1..80]]; // Vincenzo Librandi, Feb 14 2013

Table[Ceiling[3*n + Log[3*n]], {n, 100}] (* T. D. Noe, May 21 2012 *)

A212446 Floor(2n + log(2n)).

Original entry on oeis.org

2, 5, 7, 10, 12, 14, 16, 18, 20, 22, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120, 122, 124, 126, 128

Offset: 1

Mohammad K. Azarian, May 17 2012

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000523, A062153, A102572, A212445, A212447, A212448, A212449, A212450, A212451, A212452, A212453, A212454.

[Floor(2*n + Log(2*n)): n in [1..80]]; // Vincenzo Librandi, Feb 15 2013

Table[Floor[2*n + Log[2*n]], {n, 100}] (* T. D. Noe, May 21 2012 *)

a(n)=2*n+log(2*n)\1 \\ Charles R Greathouse IV, Sep 04 2015

cp's OEIS Frontend

A081604 Number of digits in ternary representation of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

Formula

A117967 Positive part of inverse of A117966; write n in balanced ternary and then replace (-1)'s with 2's.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Python

Scheme

Formula

A102572 a(n) = floor(log_4(n)).

Views

Author

Keywords

Crossrefs

Programs

Magma

PARI

PARI

Formula

A212445 a(n) = floor( n + log(n) ).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

A081134 Distance to nearest power of 3.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

A212454 Ceiling(5n + log(5n)).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

A212453 a(n) = ceiling(4n + log(4n)).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica