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A300948 a(1) = 2, a(2) = 1, and for any n > 2 with binary expansion Sum_{i=0..k} b_i * 2^i, a(n) = Sum_{i=0..k} b_i * 2^(a(i+1)-1). 3
2, 1, 3, 4, 6, 5, 7, 8, 10, 9, 11, 12, 14, 13, 15, 32, 34, 33, 35, 36, 38, 37, 39, 40, 42, 41, 43, 44, 46, 45, 47, 16, 18, 17, 19, 20, 22, 21, 23, 24, 26, 25, 27, 28, 30, 29, 31, 48, 50, 49, 51, 52, 54, 53, 55, 56, 58, 57, 59, 60, 62, 61, 63, 64, 66, 65, 67 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
More informally, in binary, the a(k)-th bit of a(n) is set iff the k-th bit of n is set (where the least significant bit has index 1).
For any k >= 0, the restriction of this sequence to the first A007013(k) terms is a self-inverse permutation preserving the Hamming weight; this property can be proven by induction.
This sequence is a self-inverse permutation of the natural numbers.
This sequence has infinitely many fixed points (A300950); for any k >= 0, at least one of 2^k or 2^k + a(2^k) is a fixed point.
See also A300955 and A300956 for sequences in the same vein.
LINKS
FORMULA
A000120(a(n)) = A000120(n).
a(a(n)) = n.
EXAMPLE
a(3) = a(2^1 + 2^0) = 2^(a(2)-1) + 2^(a(1)-1) = 2^0 + 2^1 = 3.
a(4) = a(2^2) = 2^(a(3)-1) = 2^2 = 4.
a(6) = a(2^2 + 2^1) = 2^(a(3)-1) + 2^(a(2)-1) = 2^2 + 2^0 = 5.
a(42) = a(2^5 + 2^3 + 2^1) = 2^(a(6)-1) + 2^(a(4)-1) + 2^(a(2)-1) = 2^4 + 2^3 + 2^0 = 25.
MATHEMATICA
a[1] = 2; a[2] = 1; a[n_] := a[n] = With[{b = IntegerDigits[n, 2] // Reverse}, Sum[If[b[[i]] == 1, 2^(a[i]-1), 0], {i, Length[b]}]]; Array[a, 100] (* Jean-François Alcover, Mar 17 2018 *)
PROG
(PARI) a(n) = if (n==1, 2, n==2, 1, my (b=Vecrev(binary(n))); sum(i=1, #b, if (b[i], 2^(a(i)-1), 0)))
CROSSREFS
Cf. A000120, A007013, A300950 (fixed points), A300955, A300956.
Sequence in context: A122514 A130077 A080412 * A098164 A158504 A293253
KEYWORD
nonn,base
AUTHOR
Rémy Sigrist, Mar 16 2018
STATUS
approved

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Last modified July 18 21:02 EDT 2024. Contains 374388 sequences. (Running on oeis4.)