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Lexicographically earliest sequence of distinct positive terms such that for any n > 0, the binary digits of a(n) appear in order but not necessarily as consecutive digits in the binary representation of the n-th prime number.
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%I #13 May 03 2018 11:21:28

%S 1,3,2,7,5,6,4,9,11,13,15,8,10,19,23,12,14,29,16,17,18,31,20,21,24,25,

%T 27,26,22,28,63,32,33,34,35,37,30,40,39,38,41,42,47,48,49,51,43,55,56,

%U 50,52,59,57,61,64,65,45,67,36,44,69,53,73,71,46,62,75

%N Lexicographically earliest sequence of distinct positive terms such that for any n > 0, the binary digits of a(n) appear in order but not necessarily as consecutive digits in the binary representation of the n-th prime number.

%C This sequence is a permutation of the natural numbers with inverse A303076.

%C This sequence has similarities with A286417; there binary digits have to be consecutive, here not.

%H Rémy Sigrist, <a href="/A303075/b303075.txt">Table of n, a(n) for n = 1..10000</a>

%H Rémy Sigrist, <a href="/A303075/a303075.gp.txt">PARI program for A303075</a>

%H <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>

%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>

%F a(n) <= A000040(n).

%e The first terms, alongside the binary representations of the n-th prime and of a(n), are:

%e n a(n) bin(p_n) bin(a(n))

%e -- ---- -------- ---------

%e 1 1 10 1_

%e 2 3 11 11

%e 3 2 101 10_

%e 4 7 111 111

%e 5 5 1011 101_

%e 6 6 1101 110_

%e 7 4 10001 100__

%e 8 9 10011 1001_

%e 9 11 10111 1011_

%e 10 13 11101 11_01

%e 11 15 11111 1111_

%e 12 8 100101 100_0_

%e 13 10 101001 1010__

%e 14 19 101011 10_011

%e 15 23 101111 10111_

%o (PARI) See Links section.

%Y Cf. A000040, A286417, A303076 (inverse).

%K nonn,base,look

%O 1,2

%A _Rémy Sigrist_, Apr 18 2018