A062153 -id:A062153

A081604

Number of digits in ternary representation of n.

+10
37

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

COMMENTS

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

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Eric Weisstein's World of Mathematics, Ternary.

FORMULA

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

EXAMPLE

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

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

MAPLE

A081604 := proc(n)

max(1, 1+ilog[3](n)) ;

end proc: # R. J. Mathar, Jul 12 2016

MATHEMATICA

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 *)

PROG

(Haskell)

a081604 n = if n < 3 then 1 else a081604 (div n 3) + 1

-- Reinhard Zumkeller, Sep 05 2014, Feb 21 2013

CROSSREFS

Cf. A003137, A007089, A030341, A049803, A054635, A062153, A070939, A080342.

Cf. A062756, A077267, A081603.

Cf. A134021, A134025, A134026, A246435.

KEYWORD

nonn,base

AUTHOR

Reinhard Zumkeller, Mar 23 2003

STATUS

approved

A117967

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

+10
24

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

REFERENCES

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

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..9841

Ken Levasseur, The Balanced Ternary Number System

FORMULA

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

EXAMPLE

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

MAPLE

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

MATHEMATICA

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 *)

PROG

(Scheme)

(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

(Python)

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

CROSSREFS

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

KEYWORD

base,nonn,look

AUTHOR

Franklin T. Adams-Watters, Apr 05 2006

STATUS

approved

A102572

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

+10
14

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,16

LINKS

Table of n, a(n) for n=1..105.

FORMULA

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

PROG

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

(PARI) 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

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

CROSSREFS

Cf. A000523, A062153.

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Dec 23 2006

STATUS

approved

A212445

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

+10
11

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Complement of A045650. - Michel Marcus, Jun 30 2015

LINKS

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

MATHEMATICA

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

PROG

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

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

CROSSREFS

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

KEYWORD

nonn,easy

AUTHOR

Mohammad K. Azarian, May 17 2012

STATUS

approved

A081134

Distance to nearest power of 3.

+10
10

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,5

LINKS

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

FORMULA

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)

EXAMPLE

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

MAPLE

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

MATHEMATICA

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

PROG

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

(Python)

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

CROSSREFS

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

KEYWORD

easy,nonn

AUTHOR

Klaus Brockhaus, Mar 08 2003

STATUS

approved

A212454

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

+10
10

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

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

MATHEMATICA

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

PROG

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

CROSSREFS

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

KEYWORD

nonn,easy

AUTHOR

Mohammad K. Azarian, May 17 2012

STATUS

approved

A212453

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

+10
8

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

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

MATHEMATICA

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

PROG

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

CROSSREFS

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

KEYWORD

nonn,easy

AUTHOR

Mohammad K. Azarian, May 17 2012

STATUS

approved

A212451

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

+10
7

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

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

MATHEMATICA

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

PROG

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

CROSSREFS

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

KEYWORD

nonn,easy

AUTHOR

Mohammad K. Azarian, May 17 2012

STATUS

approved

A212452

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

+10
7

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

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

MATHEMATICA

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

PROG

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

CROSSREFS

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

KEYWORD

nonn,easy

AUTHOR

Mohammad K. Azarian, May 17 2012

STATUS

approved

A212446

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

+10
6

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

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

MATHEMATICA

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

PROG

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

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

CROSSREFS

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

KEYWORD

nonn,easy

AUTHOR

Mohammad K. Azarian, May 17 2012

STATUS

approved