cp's OEIS Frontend

A294523 Lexicographically earliest sequence of positive terms, such that, for any n > 0, the binary expansion of n, say of size k+1, is (1, a(n) mod 2, a^2(n) mod 2, ..., a^k(n) mod 2) (where a^i denotes the i-th iterate of the sequence).

%I A294523 #19 Nov 03 2017 22:24:01
%S A294523 1,2,1,2,6,5,1,2,10,6,14,9,5,13,1,2,18,10,22,12,6,14,30,17,9,5,11,25,
%T A294523 13,29,1,2,34,18,38,20,10,22,46,24,12,6,54,28,14,30,62,33,17,9,19,41,
%U A294523 5,11,23,49,25,13,27,57,29,61,1,2,66,34,70,36,18,38,78
%N A294523 Lexicographically earliest sequence of positive terms, such that, for any n > 0, the binary expansion of n, say of size k+1, is (1, a(n) mod 2, a^2(n) mod 2, ..., a^k(n) mod 2) (where a^i denotes the i-th iterate of the sequence).
%C A294523 More informally, the parity of the iterate of the sequence at n gives the binary expansion of n (beyond the leading 1).
%C A294523 Apparently, iterating the sequence always leads to one of these three loops:
%C A294523 - the fixed point (1) iff we start from 2^k-1 for some k > 0,
%C A294523 - the fixed point (2) iff we start from 2^k for some k > 0,
%C A294523 - or (5, 6) for any other starting value.
%C A294523 a(n) is even iff n belongs to A004754.
%C A294523 a(n) is odd iff n belongs to A004760.
%C A294523 If a(n) > n then a(n) = A080541(n).
%C A294523 If n < 2^k then a(n) < 2^k.
%C A294523 Apparently, if a(n) > 2, then A054429(a(n)) = a(A054429(n)); this accounts for the symmetry of the part connected to the loop (5,6) in the oriented graph of this sequence.
%H A294523 Rémy Sigrist, <a href="/A294523/b294523.txt">Table of n, a(n) for n = 1..8192</a>
%H A294523 Rémy Sigrist, <a href="/A294523/a294523.png">Representation of the first 32 terms as an oriented graph (n -> a(n))</a>
%H A294523 Rémy Sigrist, <a href="/A294523/a294523_1.png">Representation of the first 64 terms as an oriented graph (n -> a(n))</a>
%H A294523 Rémy Sigrist, <a href="/A294523/a294523.gp.txt">PARI program for A294523</a>
%F A294523 a(n) = 1 iff n = A000225(k) for some k > 0.
%F A294523 a(n) = 2 iff n = A000079(k) for some k > 0.
%F A294523 a(n) = 5 iff n = A081254(k) for some k > 2.
%F A294523 a(n) = 6 iff n = A000975(k) for some k > 2.
%F A294523 a(n) = 10 iff n = A081253(k) for some k > 2.
%F A294523 a(n) = 12 iff n = A266613(k) for some k > 3.
%F A294523 a(n) = 13 iff n = A052997(k) for some k > 2.
%F A294523 a(n) = 14 iff n = A266721(k) for some k > 2.
%F A294523 a(n) = 18 iff n = A267045(k) for some k > 3.
%F A294523 a(n) = 54 iff n = A266248(k) for some k > 4.
%F A294523 These formulas come from the fact that each sequence on the right side, say f, eventually satisfies: f(n) = floor(f(n+1)/2), and f(n) and f(n+2) have the same parity.
%e A294523 For n=11:
%e A294523 - the binary representation of 11 is (1,0,1,1),
%e A294523 - a(11) = 14 has parity 0,
%e A294523 - a(14) = 13 has parity 1,
%e A294523 - a(13) = 5 has parity 1,
%e A294523 - we find the binary digits of 11 beyond the initial 1, in order: 0, 1, 1.
%e A294523 See also representations of first terms in Links section.
%o A294523 (PARI) See Links section.
%Y A294523 Cf. A000079, A000225, A000975, A004754, A004760, A052997, A054429, A080541, A081253, A081254, A266248, A266613, A266721, A267045.
%K A294523 nonn,base,look
%O A294523 1,2
%A A294523 _Rémy Sigrist_, Nov 01 2017