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A007378
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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)
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11
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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
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OFFSET
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2,1
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COMMENTS
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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
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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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
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
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MAPLE
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a := proc(n) option remember; if n < 4 then n+1 else a(iquo(n, 2)) + a(iquo(n+1, 2)) fi end:
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MATHEMATICA
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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 *)
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PROG
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(Python)
from functools import cache
@cache
def a(n): return n+1 if n < 4 else a(n//2) + a((n+1)//2)
(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
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CROSSREFS
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KEYWORD
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nonn,easy,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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