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%I #15 Feb 01 2019 05:04:06
%S 1,2,3,6,4,5,20,8,9,72,16,7,14,21,78,32,11,352,64,10,40,15,24,12,30,
%T 35,390,48,96,51,102,60,13,832,117,144,18,168,42,28,39,180,56,84,63,
%U 70,780,120,26,128,19,504,36,288,126,45,112,151,896,156,720,224
%N Lexicographically earliest sequence of distinct positive terms such that for any n > 0 and any k > 0, floor((2^k) / a(n)) AND floor((2^k) / a(n+1)) = 0 (where AND denotes the bitwise AND operator).
%C In other words, for any n > 0, the binary expansions of 1/a(n) and of 1/a(n+1) have no common one bit; in this sense, this sequence is similar to A109812.
%C This sequence is a permutation of the natural numbers, with inverse A306233 (we can first prove that all the powers of 2 appear in the sequence and then that every natural number appear in the sequence).
%H Rémy Sigrist, <a href="/A306231/b306231.txt">Table of n, a(n) for n = 1..2000</a>
%H Rémy Sigrist, <a href="/A306231/a306231.gp.txt">PARI program for A306231</a>
%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F For any n > 0, if A000120(a(n)) <> 1 and A000120(a(n+1)) <> 1, then gcd(A007733(a(n)), A007733(a(n+1))) > 1.
%e The first terms, alongside A007733(a(n)) and the binary representation of 1/a(n) with periodic part in parentheses, are:
%e n a(n) period bin(1/a(n))
%e -- ---- ------ -------------------
%e 1 1 1 1.(0)
%e 2 2 1 0.1(0)
%e 3 3 2 0.(01)
%e 4 6 2 0.0(01)
%e 5 4 1 0.01(0)
%e 6 5 4 0.(0011)
%e 7 20 4 0.00(0011)
%e 8 8 1 0.001(0)
%e 9 9 6 0.(000111)
%e 10 72 6 0.000(000111)
%e 11 16 1 0.0001(0)
%e 12 7 3 0.(001)
%e 13 14 3 0.0(001)
%e 14 21 6 0.(000011)
%e 15 78 12 0.0(000001101001)
%e 16 32 1 0.00001(0)
%e 17 11 10 0.(0001011101)
%e 18 352 10 0.00000(0001011101)
%e 19 64 1 0.000001(0)
%e 20 10 4 0.0(0011)
%o (PARI) See Links section.
%Y Cf. A000120, A007733, A109812, A306233 (inverse).
%K nonn,base
%O 1,2
%A _Rémy Sigrist_, Jan 30 2019
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