A102572 -id:A102572 - cp's OEIS frontend

A062153 a(n) = floor(log_3(n)).

0, 0, 1, 1, 1, 1, 1, 1, 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, 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

Offset: 1

Henry Bottomley, Jun 06 2001

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A000244, A000523, A007089, A102572.

[ Ilog(3,n) : n in [1..100] ]; // N. J. A. Sloane, Dec 23 2006

Floor[Log[3,Range[120]]]  (* Harvey P. Dale, Apr 30 2011 *)

a(n)=logint(n,3) \\ Charles R Greathouse IV, Nov 15 2022

a(n) = (number of digits of n when written in base 3) - 1.
a(n) = if n > 2 then a(floor(n / 3)) + 1 else 0. - Reinhard Zumkeller, Oct 29 2001
G.f.: (1/(1 - x))*Sum_{k>=1} x^(3^k). - Ilya Gutkovskiy, Jan 08 2017

A258996 Permutation of the positive integers: this permutation transforms the enumeration system of positive irreducible fractions A002487/A002487' (Calkin-Wilf) into the enumeration system A162911/A162912 (Drib), and vice versa.

Original entry on oeis.org

1, 2, 3, 6, 7, 4, 5, 10, 11, 8, 9, 14, 15, 12, 13, 26, 27, 24, 25, 30, 31, 28, 29, 18, 19, 16, 17, 22, 23, 20, 21, 42, 43, 40, 41, 46, 47, 44, 45, 34, 35, 32, 33, 38, 39, 36, 37, 58, 59, 56, 57, 62, 63, 60, 61, 50, 51, 48, 49, 54, 55, 52, 53

Offset: 1

Yosu Yurramendi, Jun 16 2015

As A258746 the permutation is self-inverse. Except for fixed points 1, 2, 3 it consists completely of 2-cycles: (4,6), (5,7), (8,10), (9,11), (12,14), (13,15), (16,26), (17,27), ..., (21,31), ..., (32,42), ... . - Yosu Yurramendi, Mar 31 2016

When terms of sequence |n - a(n)|/2 (n > 3) are considered only once, and they are sorted in increasing order, A147992 is obtained. - Yosu Yurramendi, Apr 05 2016

Yosu Yurramendi, Table of n, a(n) for n = 1..16383
Index entries for sequences that are permutations of the natural numbers

Cf. A092569, A117120, A258746. Similar R-programs: A332769, A284447.

maxlevel <- 5 # by choice
a <- 1
for(m in 0:maxlevel) for(k in 0:(2^m-1)){
  a[2^(m+1) + 2*k    ] = 2*a[2^(m+1) - 1 - k]
  a[2^(m+1) + 2*k + 1] = 2*a[2^(m+1) - 1 - k] + 1}
a

# Given n, compute a(n) by taking into account the binary representation of n
maxblock <- 7 # by choice
a <- 1:3
for(n in 4:2^maxblock){
  ones <- which(as.integer(intToBits(n)) == 1)
  nbit <- as.integer(intToBits(n))[1:tail(ones, n = 1)]
  anbit <- nbit
  anbit[seq(2, length(anbit) - 1, 2)] <- 1 - anbit[seq(2, length(anbit) - 1, 2)]
  a <- c(a, sum(anbit*2^(0:(length(anbit) - 1))))
}
a
# Yosu Yurramendi, Mar 30 2021

a(1) = 1, a(2) = 2, a(3) = 3. For n = 2^m + k, m > 1, 0 <= k < 2^m. If m is even, then a(2^(m+1)+k) = a(2^m + k) + 2^m and a(2^(m+1) + 2^m+k) = a(2^m+k) + 2^(m+1). If m is odd, then a(2^(m+1) + k) = a(2^m+k) + 2^(m+1) and a(2^(m+1) + 2^m+k) = a(2^m+k) + 2^m.
From Yosu Yurramendi, Mar 23 2017: (Start)
A258746(a(n)) = a(A258746(n)), n > 0.
A092569(a(n)) = a(A092569(n)), n > 0.
A117120(a(n)) = a(A117120(n)), n > 0;
A065190(a(n)) = a(A065190(n)), n > 0;
A054429(a(n)) = a(A054429(n)), n > 0;
A063946(a(n)) = a(A063946(n)), n > 0. (End)
a(1) = 1, for m >= 0  and 0 <= k < 2^m, a(2^(m+1) + 2*k) = 2*a(2^(m+1) - 1 - k), a(2^(m+1) + 2*k + 1) = 2*a(2^(m+1) - 1 - k) + 1. - Yosu Yurramendi, May 23 2020
a(n) = A020988(A102572(n)) XOR n. - Alan Michael Gómez Calderón, Mar 11 2025

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

A115304 a(n) = n if n < 4, otherwise 4*a(floor(n/4)) + 3 - n mod 4.

Original entry on oeis.org

1, 2, 3, 7, 6, 5, 4, 11, 10, 9, 8, 15, 14, 13, 12, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 63, 62, 61, 60, 59, 58, 57, 56, 55, 54, 53, 52, 51, 50, 49, 48, 127, 126, 125, 124, 123, 122, 121

Offset: 1

Reinhard Zumkeller, Jan 20 2006

Self-inverse permutation of the natural numbers.

If n is written in base-4 representation, then a(n) is the value after replacing all digits d (except for the leading one) by 3-d.

Cf. A054429, A102572, A115303, A115305, A115306, A115307, A115308, A115309, A106649.

A115304[n_] := FromDigits[MapAt[3 - # &, IntegerDigits[n, 4], 2;;], 4];
Array[A115304, 100] (* Paolo Xausa, May 20 2024 *)

def A115304(n): # according to Formula by Alan Michael Gómez Calderón
    m = 4**floor(log(n,4))-1
    return n^m # A.H.M. Smeets, Apr 01 2025

a(n) = A115310(n+2,3).
a(n) = n XOR (4^A102572(n) - 1). - Alan Michael Gómez Calderón, Mar 27 2025
a(a(n)) = n. - A.H.M. Smeets, Apr 01 2025

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

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

Original entry on oeis.org

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

cp's OEIS Frontend

A062153 a(n) = floor(log_3(n)).

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A258996 Permutation of the positive integers: this permutation transforms the enumeration system of positive irreducible fractions A002487/A002487' (Calkin-Wilf) into the enumeration system A162911/A162912 (Drib), and vice versa.

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

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A115304 a(n) = n if n < 4, otherwise 4*a(floor(n/4)) + 3 - n mod 4.

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Python

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

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Magma

Mathematica

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

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Magma

Mathematica

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

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Magma

Mathematica

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

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Magma