A007378 a(n), for n >= 2, is smallest positive integer which is consistent with sequence being monotonically increasing and satisfying a(a(n)) = 2n. (Formerly M2317)

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

2,1

Comments

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

References

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

Links

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

Formula

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

Maple

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

Mathematica

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 *)

Prog

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

Crossrefs

Cf. A003605. Equals A080653 + 2.

This sequence, A079905, A080637 and A080653 are all essentially the same.

Cf. A088720, A042965.

Sequence in context: A022846 A083922 A039042 * A274829 A087758 A227019

Adjacent sequences: A007375 A007376 A007377 * A007379 A007380 A007381

Keyword

nonn,easy,nice

Author

Colin Mallows

Extensions

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

Status

approved