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Lexicographically earliest sequence of distinct positive integers such that for any n > 0, if 2^(d-1) appears in the binary expansion of a(n) then d divides n.
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%I #6 Oct 02 2023 13:49:00

%S 1,2,4,3,16,5,64,8,256,17,1024,6,4096,65,20,9,65536,7,262144,10,68,

%T 1025,4194304,11,16777216,4097,257,66,268435456,18,1073741824,128,

%U 1028,65537,80,12,68719476736,262145,4100,19,1099511627776,32,4398046511104,1026,21

%N Lexicographically earliest sequence of distinct positive integers such that for any n > 0, if 2^(d-1) appears in the binary expansion of a(n) then d divides n.

%C In other words, the binary expansion of a(n) encodes a subset of the divisors of n.

%C This sequence is a permutation of the positive integers with inverse A366028.

%H Rémy Sigrist, <a href="/A366027/a366027.gp.txt">PARI program</a>

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

%F a(p) = 2^(p-1) for any prime number p.

%F a(2*p) = 2^(p-1) + 1 for any prime number p.

%e The first terms, alongside their binary expansion and the corresponding divisors d, are:

%e n a(n) bin(a(n)) Corresponding divisors

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

%e 1 1 1 {1}

%e 2 2 10 {2}

%e 3 4 100 {3}

%e 4 3 11 {2, 1}

%e 5 16 10000 {5}

%e 6 5 101 {3, 1}

%e 7 64 1000000 {7}

%e 8 8 1000 {4}

%e 9 256 100000000 {9}

%e 10 17 10001 {5, 1}

%e 11 1024 10000000000 {11}

%e 12 6 110 {3, 2}

%e 13 4096 1000000000000 {13}

%e 14 65 1000001 {7, 1}

%e 15 20 10100 {5, 3}

%e 16 9 1001 {4, 1}

%e 17 65536 10000000000000000 {17}

%e 18 7 111 {3, 2, 1}

%o (PARI) See Links section.

%Y Cf. A048793, A271410, A366028 (inverse).

%K nonn,base

%O 1,2

%A _Rémy Sigrist_, Sep 26 2023