A003605 -id:A003605 - cp's OEIS frontend

A079000 a(n) is taken to be the smallest positive integer greater than a(n-1) which is consistent with the condition "n is a member of the sequence if and only if a(n) is odd".

1, 4, 6, 7, 8, 9, 11, 13, 15, 16, 17, 18, 19, 20, 21, 23, 25, 27, 29, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 95, 97

Offset: 1

Matthew Vandermast, Feb 01 2003

a(a(n)) = 2n + 3 for n>1.

			a(2) cannot be 2 because 2 is even; it cannot be 3 because that would require 2 to be a member of the sequence. Hence a(2)=4 and the next odd member of the sequence is the fourth member.

Hsien-Kuei Hwang, S Janson, TH 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; http://140.109.74.92/hk/wp-content/files/2016/12/aat-hhrr-1.pdf. Also 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
N. J. A. Sloane, Seven Staggering Sequences, in Homage to a Pied Puzzler, E. Pegg Jr., A. H. Schoen and T. Rodgers (editors), A. K. Peters, Wellesley, MA, 2009, pp. 93-110.

N. J. A. Sloane, Table of n, a(n) for n = 1..10000
B. Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence, J. Integer Seqs., Vol. 6 (2003), #03.2.2.
B. Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence, arXiv:math/0305308 [math.NT], 2003.
N. J. A. Sloane, Seven Staggering Sequences.
Index entries for sequences of the a(a(n)) = 2n family

Cf. A079250-A079259, A079313, A079325, A064437, A003605, A079352, A079358.
Cf. also A080596, A080731, A080752, A121053, A080032.
Partial sums give A080566. Differences give A079948.

Digits := 50; A079000 := proc(n) local k,j; if n<=2 then n^2; else k := floor(evalf(log( (n+3)/6 )/log(2)) ); j := n-(9*2^k-3); 12*2^k-3+3*j/2 +abs(j)/2; fi; end;
A002264 := n->floor(n/3): A079944 := n->floor(log[2](4*(n+2)/3))-floor(log[2](n+2)): A000523 := n->floor(log[2](n)): f := n->A079944(A002264(n-4)): g := n->A000523(A002264(n+2)/2): A079000 := proc(n) if n>3 then RETURN(simplify(3*n+3-3*2^g(n)+(-1)^f(n)*(9*2^g(n)-n-3))/2) else if n>0 then RETURN([1,4,6][n]) else RETURN(0) fi fi: end;

a[1] = 1; a[n_] := (k = Floor[Log[2, (n+3)/6]]; j = n-(9*2^k - 3); 12*2^k-3 + 3*j/2 + Abs[j]/2); Table[a[n], {n, 1, 71}] (* Jean-François Alcover, May 21 2012, after Maple *)

a(1) = 1, a(2) = 4, then a(9*2^k-3+j) = 12*2^k-3+3*j/2+|j|/2 for k>=0, -3*2^k <= j <= 3*2^k. Also a(3n) = 3*b(n/3), a(3n+1) = 2*b(n)+b(n+1), a(3n+2) = b(n)+2*b(n+1) for n>=2, where b = A079905. - N. J. A. Sloane and Benoit Cloitre, Feb 20 2003
a(n+1) - 2*a(n) + a(n-1) = 1 for n = 9*2^k - 3, k>=0, = -1 for n = 2 and 3*2^k-3, k>=1 and = 0 otherwise.
a(n) = (3*n + 3 - 3*2^g(n) + (-1)^f(n)*(9*2^g(n) - n - 3))/2 for n>3, f(n) = A079944(A002264(n-4)) and g(n) = A000523(A002264(n+2)/2). - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Feb 23 2003
Also a(n) = n + 3*2^A000523(A002264(n+2)/2)*(1 - 3*A080584(n-4)) + A080584(n-4)*(n+3) for n>3, where A080584(n)=A079944(A002264(n)). - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Feb 24 2003

A007378 a(n), for n >= 2, is smallest positive integer which is consistent with sequence being monotonically increasing and satisfying a(a(n)) = 2n.

Original entry on oeis.org

3, 4, 6, 7, 8, 10, 12, 13, 14, 15, 16, 18, 20, 22, 24, 25, 26, 27, 28, 29, 30, 31, 32, 34, 36, 38, 40, 42, 44, 46, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 97, 98, 99, 100, 101, 102, 103

Offset: 2

Colin Mallows

This is the unique monotonic sequence {a(n)}, n>=2, satisfying a(a(n)) = 2n.
May also be defined by: a(n), n=2,3,4,..., is smallest positive integer greater than a(n-1) which is consistent with the condition "n is a member of the sequence if and only if a(n) is an even number >= 4". - N. J. A. Sloane, Feb 23 2003
A monotone sequence satisfying a^(k+1)(n) = mn is unique if m=2, k >= 0 or if (k,m) = (1,3). See A088720. - Colin Mallows, Oct 16 2003
Numbers (greater than 2) whose binary representation starts with "11" or ends with "0". - Franklin T. Adams-Watters, Jan 17 2006
Lower density 2/3, upper density 3/4. - Charles R Greathouse IV, Dec 14 2022

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 2..10000
J.-P. Allouche, N. Rampersad and J. Shallit, On integer sequences whose first iterates are linear, Aequationes Math. 69 (2005), 114-127.
J.-P. Allouche and J. Shallit, The Ring of k-regular Sequences, II
J.-P. Allouche and J. Shallit, The ring of k-regular sequences, II, Theoret. Computer Sci., 307 (2003), 3-29.
Benoit Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence, J. Integer Seqs., Vol. 6 (2003), #03.2.2.
Benoit Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence, arXiv:math/0305308 [math.NT], 2003.
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.
Jeffrey Shallit, k-regular Sequences
Ralf Stephan, Some divide-and-conquer sequences ...
Ralf Stephan, Table of generating functions
Index entries for sequences of the a(a(n)) = 2n family

Cf. A003605. Equals A080653 + 2.
This sequence, A079905, A080637 and A080653 are all essentially the same.
Cf. A088720, A042965.

a := proc(n) option remember; if n < 4 then n+1 else a(iquo(n,2)) + a(iquo(n+1,2)) fi end:
seq(a(n), n = 2..70); # Peter Bala, Aug 03 2022

max = 70; f[x_] := -x/(1-x) + x/(1-x)^2*(2 + Sum[ x^(2^k + 2^(k+1)) - x^2^(k+1) , {k, 0, Ceiling[Log[2, max]]}]); Drop[ CoefficientList[ Series[f[x], {x, 0, max + 1}], x], 2](* Jean-François Alcover, May 16 2012, from g.f. *)
a[2]=3; a[3]=4; a[n_?OddQ] := a[n] = a[(n-1)/2+1] + a[(n-1)/2]; a[n_?EvenQ] := a[n] = 2a[n/2]; Table[a[n], {n, 2, 71}] (* Jean-François Alcover, Jun 26 2012, after Vladeta Jovovic *)

a(n) = my(s=logint(n,2)-1); if(bittest(n,s), n<<1 - 2<Kevin Ryde, Aug 08 2022

from functools import cache
@cache
def a(n): return n+1 if n < 4 else a(n//2) + a((n+1)//2)
print([a(n) for n in range(2, 72)]) # Michael S. Branicky, Aug 04 2022

a(2^i + j) = 3*2^(i-1) + j, 0<=j<2^(i-1); a(3*2^(i-1) + j) = 2^(i+1) + 2*j, 0<=j<2^(i-1).
a(3*2^k + j) = 4*2^k + 3j/2 + |j|/2, k>=0, -2^k <= j < 2^k. - N. J. A. Sloane, Feb 23 2003
a(2*n+1) = a(n+1)+a(n), a(2*n) = 2*a(n). a(n) = n+A060973(n). - Vladeta Jovovic, Mar 01 2003
G.f.: -x/(1-x) + x/(1-x)^2 * (2 + sum(k>=0, t^2(t-1), t=x^2^k)). - Ralf Stephan, Sep 12 2003

More terms from Matthew Vandermast and Vladeta Jovovic, Mar 01 2003

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)

A080637 a(n) is the smallest positive integer which is consistent with the sequence being monotonically increasing and satisfying a(1)=2, a(a(n)) = 2n+1 for n > 1.

Original entry on oeis.org

2, 3, 5, 6, 7, 9, 11, 12, 13, 14, 15, 17, 19, 21, 23, 24, 25, 26, 27, 28, 29, 30, 31, 33, 35, 37, 39, 41, 43, 45, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 96, 97, 98, 99, 100, 101, 102

Offset: 1

N. J. A. Sloane and Benoit Cloitre, Feb 28 2003

Sequence is the unique monotonic sequence satisfying a(a(n)) = 2n+1.

Except for the first term, numbers (greater than 2) whose binary representation starts with 11 or ends with 1. - Yifan Xie, May 26 2022

			From _Yifan Xie_, May 02 2022: (Start)
a(8) = 12 because 2*2^2 <= 8 < 3*2^2, hence a(8) = 8 + 2^2 = 12;
a(13) = 19 because 3*2^2 <= 13 < 4*2^2, hence a(13) = 2*(13 - 2^2) + 1 = 19. (End)

Yifan Xie, Table of n, a(n) for n = 1..10000
Benoit Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence, J. Integer Seqs., Vol. 6 (2003), #03.2.2.
Benoit Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence, arXiv:math/0305308 [math.NT], 2003.
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 of the a(a(n)) = 2n family

Except for first term, same as A079905. Cf. A079000.
A007378, A079905, A080637, A080653 are all essentially the same sequence.
Equals A007378(n+1)-1. First differences give A079882.
Unique monotonic sequence of positive integers satisfying a(a(n)) = k*(n-1) + 3: this sequence (k=2), A003605 (k=3), A353651 (k=4), A353652 (k=5), A353653 (k=6).

t := []; for k from 0 to 6 do for j from -2^k to 2^k-1 do t := [op(t), 4*2^k - 1 + 3*j/2 + abs(j)/2]; od: od: t;

b[n_] := b[n] = If[n<4, n+1, If[OddQ[n], b[(n-1)/2+1]+b[(n-1)/2], 2b[n/2]]];
a[n_] := b[n+1]-1;
a /@ Range[70] (* Jean-François Alcover, Oct 31 2019 *)

a(3*2^k - 1 + j) = 4*2^k - 1 + 3*j/2 + |j|/2 for k >= 0, -2^k <= j < 2^k.
a(2n+1) = 2*a(n) + 1, a(2n) = a(n) + a(n-1) + 1.
From Yifan Xie, May 02 2022: (Start)
For n in the range 2*2^i <= n < 3*2^i, for i >= 0:
  a(n) = n + 2^i.
  a(n) = 1 + a(n-1).
Otherwise, for n in the range 3*2^i <= n < 4*2^i, for i >= 0:
  a(n) = 2*(n - 2^i) + 1.
  a(n) = 2 + a(n-1). (End)

A080720 a(0) = 5; for n>0, a(n) is taken to be the smallest positive integer greater than a(n-1) which is consistent with the condition "n is a member of the sequence if and only if a(n) is a multiple of 3".

Original entry on oeis.org

5, 7, 8, 10, 11, 12, 13, 15, 18, 19, 21, 24, 27, 30, 31, 33, 34, 35, 36, 39, 40, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 54, 55, 57, 60, 63, 66, 67, 68, 69, 72, 73, 75, 78, 81, 84, 87, 90, 93, 96, 99, 102, 103, 104, 105, 108, 109, 111, 112, 113, 114, 115, 116, 117, 118, 119

Offset: 0

N. J. A. Sloane, Mar 05 2003

B. Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence, J. Integer Seqs., Vol. 6 (2003), #03.2.2.
B. Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence (math.NT/0305308)
Index entries for sequences of the a(a(n)) = 2n family

Cf. A079000, A003605, A079253, A080711, A080710.

{a=5; m=[5]; for(n=1,66,print1(a,","); a=a+1; if(m[1]==n, while(a%3>0,a++); m=if(length(m)==1,[],vecextract(m,"2..")),if(a%3==0,a++)); m=concat(m,a))}

a(a(n)) = 3*(n+4).

More terms and PARI code from Klaus Brockhaus, Mar 07 2003

A353651 Unique monotonic sequence of positive integers satisfying a(a(n)) = k*(n-1) + 3, where k = 4.

Original entry on oeis.org

2, 3, 7, 8, 9, 10, 11, 15, 19, 23, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 47, 51, 55, 59, 63, 67, 71, 75, 79, 83, 87, 91, 95, 99, 103, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125

Offset: 1

Yifan Xie, May 02 2022

Numbers m such that the base-4 representation of (3*m-1) starts with 11 or 12 or 13 or ends with 0.

First differences give a run of 4^i 1's followed by a run of 4^i 4's, for i = 0, 1, 2, ...

			a(6) = 10 because (2*4^1 + 1)/3 < 6 <= (5*4^1 + 1)/3, hence a(6) = 6 + 4^1 = 10;
a(9) = 19 because (5*4^1 + 1)/3 < 9 <= (8*4^1 + 1)/3, hence a(9) = 4*(9 - 4^1) - 1 = 19.

For other values of k: A080637 (k=2), A003605 (k=3), this sequence (k=4), A353652 (k=5), A353653 (k=6).

isA353651 := proc(n)
    if modp(n,4) = 3 then
        true;
    else
        b4 := convert(3*n-1,base,4) ;
        if op(-1,b4) = 1 and op(-2,b4) <> 0  then
            true ;
        else
            false;
        end if;
    end if;
end proc:
for n from 2 to 122 do
    if isA353651(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Jul 05 2022

a(n) = my(n3=3*n, s=logint(n3>>1, 4)<<1); if(n3>>s < 5, n + 1<Kevin Ryde, Apr 15 2022
(C++)
/* program used to generate the b-file */
#include
using namespace std;
int main(){
    int cnt1=1, flag=0, cnt2=1, a=2;
    for(int n=1; n<=10000; n++) {
        cout<

For n in the range (2*4^i + 1)/3 < n <= (5*4^i + 1)/3, for i >= 0:
  a(n) = n + 4^i.
  a(n) = 1 + a(n-1).
Otherwise, for n in the range (5*4^i + 1)/3 < n <= (8*4^i + 1)/3, for i >= 0:
  a(n) = 4*(n - 4^i) - 1.
  a(n) = 4 + a(n-1).

A353652 Unique monotonic sequence of positive integers satisfying a(a(n)) = k*(n-1) + 3, where k = 5.

Original entry on oeis.org

2, 3, 8, 9, 10, 11, 12, 13, 18, 23, 28, 33, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 68, 73, 78, 83, 88, 93, 98, 103, 108, 113, 118, 123, 128, 133, 138, 143, 148, 153, 158, 163, 168, 173, 178, 183, 188

Offset: 1

Yifan Xie, Jul 15 2022

Numbers m such that the base-5 representation of (2*m-1) starts with 3 or 4 or ends with 0.

First differences give a run of 5^i 1's followed by a run of 5^i 5's, for i >= 0.

			a(7) = 12 because (5^1 + 1)/2 <= 7 < (3*5^1 + 1)/2, hence a(7) = 7 + 5^1 = 12;
a(11) = 28 because (3*5^1 + 1)/2 <= 11 < (5*5^1 + 1)/2, hence a(11) = 5*(11 - 5^1) - 2 = 28.

For other values of k: A080637 (k=2), A003605 (k=3), A353651 (k=4), this sequence (k=5), A353653 (k=6).

a[n_] := Module[{n2 = 2n, p}, p = 5^Floor@Log[5, n2]; If[n2 < 3p, n+p, 5(n-p)-2]];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Sep 22 2023, after Kevin Ryde *)

a(n) = my(n2=n<<1, p=5^logint(n2, 5)); if(n2 < 3*p, n+p, 5*(n-p)-2); \\ Kevin Ryde, Apr 18 2022
(C++)
/* program used to generate the b-file */
#include
using namespace std;
int main(){
    int cnt1=1, flag=0, cnt2=1, a=2;
    for(int n=1; n<=10000; n++) {
        cout<

For n in the range (5^i + 1)/2 <= n < (3*5^i + 1)/2, for i >= 0:
  a(n) = n + 5^i.
  a(n+1) = 1 + a(n).
Otherwise, for n in the range (3*5^i + 1)/2 < n <= (5*5^i + 1)/2, for i >= 0:
  a(n) = 5*(n - 5^i) - 2.
  a(n+1) = 5 + a(n).

A353653 Unique monotonic sequence of positive integers satisfying a(a(n)) = k*(n-1) + 3, where k = 6.

Original entry on oeis.org

2, 3, 9, 10, 11, 12, 13, 14, 15, 21, 27, 33, 39, 45, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 93, 99, 105, 111, 117, 123, 129, 135, 141, 147, 153, 159, 165

Offset: 1

Yifan Xie, Jul 15 2022

Numbers m such that the base-6 representation of (5*m-3) starts with 11 or 12 or 13 or 14 or 15 or ends with 0.

First differences give a run of 6^i 1's followed by a run of 6^i 6's, for i >= 0.

			a(5) = 11 because (2*6^1 + 3)/3 <= 5 < (7*6^1 + 3)/5, hence a(5) = 5 + 6^1 = 11;
a(10) = 21 because (7*6^1 + 3)/5 <= 10 < (12*6^1 + 3)/5, hence a(10) = 6*(10 - 6^1) - 3 = 21.

For other values of k: A080637 (k=2), A003605 (k=3), A353651 (k=4), A353652 (k=5), this sequence (k=6).

okQ[m_] := With[{id = IntegerDigits[5 m - 3, 6] }, MatchQ[id[[1 ;; 2]], {1, 1}|{1, 2}|{1, 3}|{1, 4}|{1, 5}] || id[[-1]] == 0];
Join[{2}, Select[Range[3, 1000], okQ]] (* Jean-François Alcover, Sep 22 2023 *)

For n in the range (2*6^i + 3)/5 <= n < (7*6^i + 3)/5, for i >= 0:
  a(n) = n + 6^i.
  a(n+1) = 1 + a(n).
Otherwise, for n in the range (7*6^i + 3)/5 <= n < (12*6^i + 3)/5, for i >= 0:
  a(n) = 6*(n - 6^i) - 3.
  a(n+1) = 6 + a(n).

A080710 a(0) = 1; for n>0, a(n) is taken to be the smallest positive integer greater than a(n-1) which is consistent with the condition "n is a member of the sequence if and only if a(n) is a multiple of 3".

Original entry on oeis.org

1, 3, 4, 6, 9, 10, 12, 13, 14, 15, 18, 19, 21, 24, 27, 30, 31, 32, 33, 36, 37, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 51, 54, 57, 58, 59, 60, 63, 64, 66, 69, 72, 75, 78, 81, 84, 87, 90, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 105, 108, 111, 112, 113, 114, 117, 118, 120

Offset: 0

N. J. A. Sloane, Mar 05 2003

Is this the same sequence as A115837? - Andrew S. Plewe, May 08 2007

B. Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence, J. Integer Seqs., Vol. 6 (2003), #03.2.2.
B. Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence (math.NT/0305308)
Index entries for sequences of the a(a(n)) = 2n family

Cf. A003605, A079253, A080711, A080712.

Cf. A079000, A003605, A079253, A080711, A080712.

{a=1; m=[1]; for(n=1,67,print1(a,","); a=a+1; if(m[1]==n, while(a%3>0,a++); m=if(length(m)==1,[],vecextract(m,"2.."))); m=concat(m,a))}

a(a(n)) = 3*(n+1).

More terms and PARI code from Klaus Brockhaus, Mar 06 2003

A006166 a(0)=0, a(1)=a(2)=1; for n >= 1, a(3n+2) = 2a(n+1) + a(n), a(3n+1) = a(n+1) + 2a(n), a(3n) = 3a(n).

Original entry on oeis.org

0, 1, 1, 3, 3, 3, 3, 5, 7, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69

Offset: 0

N. J. A. Sloane

J. Arkin, D. C. Arney, L. S. Dewald and W. E. Ebel, Jr., Families of recursive sequences, J. Rec. Math., 22 (No. 22, 1990), 85-94.
vN. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Jean-Paul Allouche and Jeffrey Shallit, The ring of k-regular sequences, II, Theoret. Computer Sci., 307 (2003), 3-29. [Preprint.]
Hsien-Kuei Hwang, Svante Janson and Tsung-Hsi 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, Svante Janson and Tsung-Hsi Tsai, Exact and Asymptotic Solutions of a Divide-and-Conquer Recurrence Dividing at Half: Theory and Applications, ACM Transactions on Algorithms, Vol. 13, No. 4 (2017), Article #47.

a(n) + n = A003605(n). Cf. A006165, A080678, A081134.

From Peter Bala, Oct 08 2022: (Start)
a(n+2) - a(n) = 0 or 2.
a(3^k + j) = 3^k for k >= 0 and for 0 <= j <= 3^k.
a(2*3^k + j) = 3^k + 2*j for k >= 0 and for 0 <= j <= 3^k.
A081134(n) = n - a(n). (End)

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Mar 28 2003

cp's OEIS Frontend

A079000 a(n) is taken to be the smallest positive integer greater than a(n-1) which is consistent with the condition "n is a member of the sequence if and only if a(n) is odd".

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A007378 a(n), for n >= 2, is smallest positive integer which is consistent with sequence being monotonically increasing and satisfying a(a(n)) = 2n.

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A081134 Distance to nearest power of 3.

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A080637 a(n) is the smallest positive integer which is consistent with the sequence being monotonically increasing and satisfying a(1)=2, a(a(n)) = 2n+1 for n > 1.

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A080720 a(0) = 5; for n>0, a(n) is taken to be the smallest positive integer greater than a(n-1) which is consistent with the condition "n is a member of the sequence if and only if a(n) is a multiple of 3".

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A353651 Unique monotonic sequence of positive integers satisfying a(a(n)) = k*(n-1) + 3, where k = 4.

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A353652 Unique monotonic sequence of positive integers satisfying a(a(n)) = k*(n-1) + 3, where k = 5.

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