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A352808
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Lexicographically earliest sequence of distinct nonnegative integers such that for any n >= 0, a(n) AND a(floor(n/2)) = 0 (where AND denotes the bitwise AND operator).
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7
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0, 1, 2, 4, 5, 8, 3, 9, 10, 16, 6, 7, 12, 20, 18, 22, 17, 21, 11, 13, 24, 25, 32, 40, 19, 33, 34, 35, 36, 37, 41, 64, 14, 38, 42, 66, 48, 52, 50, 80, 39, 65, 68, 70, 15, 23, 67, 69, 44, 72, 26, 28, 29, 73, 76, 84, 27, 74, 82, 88, 86, 128, 30, 31, 49, 81, 89
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,3
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COMMENTS
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This sequence has connections with A109812; each term (except the first) has no 1 bit in common with some prior term.
This sequence is a permutation of the natural numbers. [Proof: The first term divisible by a given power of 2, 2^k say, is 2^k itself, and for k >= 3, it is immediately followed by the smallest missing number. Since there are infinitely many powers of 2, every number will eventually appear. - N. J. A. Sloane, May 17 2022]
An alternative and equivalent definition: a(0)=0, a(1)=1. For k >= 1, a(2*k) and a(2*k+1) are the two smallest numbers not yet in the sequence whose binary expansions have no 1's in common with the binary expansion of a(k). - N. J. A. Sloane, May 17 2022
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LINKS
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EXAMPLE
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The initial terms a(n), alongside the binary expansions of a(n) and a(floor(n/2)), are:
n a(n) bin(a(n)) bin(a(floor(n/2)))
-- ---- --------- ------------------
0 0 0 0
1 1 1 0
2 2 10 1
3 4 100 1
4 5 101 10
5 8 1000 10
6 3 11 100
7 9 1001 100
8 10 1010 101
9 16 10000 101
10 6 110 1000
11 7 111 1000
12 12 1100 11
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PROG
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(C++) See Links section.
(Python)
from itertools import count, islice
def agen(): # generator of terms
alst = [0, 1]; aset = {0, 1}; yield from alst
mink = 2
for n in count(2):
ahalf, k = alst[n//2], mink
while k in aset or k&ahalf: k += 1
alst.append(k); aset.add(k); yield k
while mink in aset: mink += 1
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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