A062153 -id:A062153 - cp's OEIS frontend

A212447 a(n) = floor(3n + log(3n)).

4, 7, 11, 14, 17, 20, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 61, 64, 67, 70, 73, 76, 79, 82, 85, 88, 91, 94, 97, 100, 103, 106, 109, 112, 115, 118, 121, 124, 127, 130, 133, 136, 139, 142, 145, 148, 151, 155, 158, 161, 164, 167, 170, 173, 176, 179

Offset: 1

Mohammad K. Azarian, May 17 2012

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

Cf. A000523, A062153, A102572, A212445, A212446, A212448, A212449, A212450, A212451, A212452, A212453, A212454.

[Floor(3*n + Log(3*n)): n in [1..80]]; // Vincenzo Librandi, Feb 15 2013

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

a(n)=3*n+log(3*n)\1 \\ Charles R Greathouse IV, Sep 04 2015

A212448 Floor(4n + log(4n)).

Original entry on oeis.org

5, 10, 14, 18, 22, 27, 31, 35, 39, 43, 47, 51, 55, 60, 64, 68, 72, 76, 80, 84, 88, 92, 96, 100, 104, 108, 112, 116, 120, 124, 128, 132, 136, 140, 144, 148, 152, 157, 161, 165, 169, 173, 177, 181, 185, 189, 193, 197, 201, 205, 209, 213, 217, 221, 225, 229

Offset: 1

Mohammad K. Azarian, May 17 2012

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

Cf. A000523, A062153, A102572, A212445, A212446, A212447, A212449, A212450, A212451, A212452, A212453, A212454.

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

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

a(n)=4*n+log(4*n)\1 \\ Charles R Greathouse IV, Sep 04 2015

A212449 Floor(5n + log(5n)).

Original entry on oeis.org

6, 12, 17, 22, 28, 33, 38, 43, 48, 53, 59, 64, 69, 74, 79, 84, 89, 94, 99, 104, 109, 114, 119, 124, 129, 134, 139, 144, 149, 155, 160, 165, 170, 175, 180, 185, 190, 195, 200, 205, 210, 215, 220, 225, 230, 235, 240, 245, 250, 255, 260, 265, 270, 275, 280, 285

Offset: 1

Mohammad K. Azarian, May 17 2012

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

Cf. A000523, A062153, A102572, A212445, A212446, A212447, A212448, A212450, A212451, A212452, A212453, A212454.

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

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

a(n)=5*n+log(5*n)\1 \\ Charles R Greathouse IV, Sep 04 2015

A212450 a(n) = ceiling(n + log(n)).

Original entry on oeis.org

1, 3, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 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, 58, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71

Offset: 1

Mohammad K. Azarian, May 17 2012

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

Cf. A000523, A062153, A102572, A212445, A212446, A212447, A212448, A212449, A212451, A212452, A212453, A212454.

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

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

A061282 Minimal number of steps to get from 0 to n by (a) adding 1 or (b) multiplying by 3. A stopping problem: begin with n and at each stage if a multiple of 3 divide by 3, otherwise subtract 1.

Original entry on oeis.org

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

Offset: 0

Henry Bottomley, Jun 06 2001

n > 0 occurs A001590(n+2) times in this sequence. - Peter Kagey, Jul 19 2015

a(n) gives the number of iterations of A260316 to reach 0. - Peter Kagey, Jul 22 2015

			a(25)=7 since 25=((0+1+1)*3+1+1)*3+1.

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Index to sequences related to the complexity of n

Cf. A001590, A007089, A053735, A056792, A062153, A260316.

Analogous sequences with a different multiplier k: A056792 (k=2), A260112 (k=4).

c i = if i `mod` 3 == 0 then i `div` 3 else i - 1
b 0 foldCount = foldCount
b sheetCount foldCount = b (c sheetCount) (foldCount + 1)
a061282 n = b n 0 -- Peter Kagey, Sep 02 2015

a:= n-> (l-> nops(l)+add(i, i=l)-1)(convert(n, base, 3)):
seq(a(n), n=0..105);  # Alois P. Heinz, Jul 16 2015

a(n)=sumdigits(n,3)+#digits(n,3)-1 \\ Charles R Greathouse IV, Jul 16 2015

a(n) = A062153(n) + A053735(n) = (number of base 3 digits of n) + (sum of base 3 digits of n)-1. a(3n) = a(n)+1, a(3n+1) = a(n)+2, a(3n+2) = a(n)+3; a(0)=0, a(1)=1, a(2)=2.

A054965 Beatty sequence for log_3(10), i.e., for 1/log_10(3); so largest exponent of 3 which produces an n-digit decimal number.

Original entry on oeis.org

2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 88, 90, 92, 94, 96, 98, 100, 102, 104, 106, 108, 111, 113, 115, 117, 119, 121, 123, 125, 127, 129, 132, 134, 136

Offset: 1

Henry Bottomley, Dec 13 2002

			log_10(3) = 0.477121... so a(11) = floor(11/0.477121...) = floor(23.0549...) = 23; 3^23 = 94143178827 is the largest 11 decimal digit power of 3.

Paolo Xausa, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Beatty Sequence.
Index entries for sequences related to Beatty sequences.

Cf. A062153, A066343, A074118, A152566.

Floor[Range[100]*Log[3, 10]] (* Paolo Xausa, Jul 11 2024 *)

a(n) = n*log(10)\log(3); \\ Michel Marcus, Aug 03 2017

a(n) = floor(n/log_10(3)) = log_3(A074118(n)) = A062153(A074118(n)).

A065335 3-exponents to represent 3-smooth numbers (A065332).

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Oct 29 2001

nonn

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

Cf. A006519, A007949, A062153, A065332, A065333, A065334.

a[n_] := If[n/2^IntegerExponent[n, 2]/3^(e = IntegerExponent[n, 3]) == 1, e, 0]; Array[a, 100] (* Amiram Eldar, Feb 21 2021 *)

a(n) = A007949(n) * A065333(n).

a(n) = log_3(n / A006519(n)), where log_3 = A062153. For k > 0 with A065332(k) > 0: A065332(k) = (2^A065334(k)) * (3^a(k)).

A105202 Irregular triangle read by rows: row n gives the word f(f(f(...(1)))) [with n applications of f], where f is the morphism 1->{1,2,1}, 2->{2,3,2}, 3->{3,1,3}.

Original entry on oeis.org

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

Offset: 0

Roger L. Bagula, Apr 09 2005

Row n contains 3^n symbols.

			From _Antti Karttunen_, Aug 12 2017: (Start)
The rows 0 .. 3 of this irregular triangle:
  1
  1;2;1
  1 2 1;2 3 2;1 2 1;
  1 2 1 2 3 2 1 2 1;2 3 2 3 1 3 2 3 2;1 2 1 2 3 2 1 2 1
(End)

Antti Karttunen, Table of n, a(n) for n = 0..9840
F. M. Dekking, Recurrent sets, Advances in Mathematics, vol. 44, no. 1 (1982), 78-104; page 96, section 4.10.

Cf. A003462, A062153, A073058, A105141.

Each row is a prefix of A105203.

f[n_] := Nest[ Flatten[ # /. {1 -> {1, 2, 1}, 2 -> {2, 3, 2}, 3 -> {3, 1, 3}}] &, {1}, n]; Flatten[ Table[ f[n], {n, 0, 4}]] (* Robert G. Wilson v, Apr 12 2005 *)

Let r = A062153(1+(2*n)) [index of the row], let c = n - A003462(r) [index of the column], then a(n) = 1 + (a(A003462(r-1)+floor(c/3)) mod 3) if n ≡ 2 mod 3, otherwise a(n) = a(A003462(r-1)+floor(c/3)). - Antti Karttunen, Aug 12 2017

More terms from Robert G. Wilson v, Apr 12 2005

A105203 Trajectory of 1 under the morphism f: 1->{1,2,1}, 2->{2,3,2}, 3->{3,1,3}.

Original entry on oeis.org

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

Offset: 0

Roger L. Bagula, Apr 09 2005

nonn

Antti Karttunen, Table of n, a(n) for n = 0..19682
F. M. Dekking, Recurrent sets, Advances in Mathematics, vol. 44, no. 1 (1982), 78-104; page 96, section 4.10.
Index entries for sequences that are fixed points of mappings

Cf. A003462, A062153, A073058, A105141, A105202.

Nest[ Flatten[ # /. {1 -> {1, 2, 1}, 2 -> {2, 3, 2}, 3 -> {3, 1, 3}}] &, {1}, 5] (* Robert G. Wilson v, Apr 12 2005 *)

(define (A105203 n) (if (zero? n) 1 (A105202 (+ n (A003462 (+ 1 (A062153 n))))))) ;; Antti Karttunen, Aug 12 2017

a(0) = 1; and for n > 1, a(n) = A105202(n+A003462(1+A062153(n))). - Antti Karttunen, Aug 12 2017

More terms from Robert G. Wilson v, Apr 12 2005

A329194 a(n) = floor(log_3(n^2)) = floor(2 log_3(n)).

Original entry on oeis.org

0, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8

Offset: 1

M. F. Hasler, Nov 07 2019

nonn

Cf. A000290 (n^2), A062153 (log_3), A329202 (log_2(n^2)), A329193 (log_2(n^3)).

Table[Floor[Log[3,n^2]],{n,120}] (* Harvey P. Dale, May 04 2025 *)

apply( A329194(n)=logint(n^2,3), [1..99])

2*A062153(n) <= a(n) = floor(log_3(n^2)) = A062153(A000290(n)).

This A329194 = A062153 o A000290.

cp's OEIS Frontend

A212447 a(n) = floor(3n + log(3n)).

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Magma

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A212448 Floor(4n + log(4n)).

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Magma

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A212449 Floor(5n + log(5n)).

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Magma

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A212450 a(n) = ceiling(n + log(n)).

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Magma

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A061282 Minimal number of steps to get from 0 to n by (a) adding 1 or (b) multiplying by 3. A stopping problem: begin with n and at each stage if a multiple of 3 divide by 3, otherwise subtract 1.

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A054965 Beatty sequence for log_3(10), i.e., for 1/log_10(3); so largest exponent of 3 which produces an n-digit decimal number.

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Mathematica

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A065335 3-exponents to represent 3-smooth numbers (A065332).

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Formula

A105202 Irregular triangle read by rows: row n gives the word f(f(f(...(1)))) [with n applications of f], where f is the morphism 1->{1,2,1}, 2->{2,3,2}, 3->{3,1,3}.

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