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A302659 Lexicographically earliest sequence of distinct positive terms such that, for any n > 0, A065368(a(n) * a(n+1)) = 0.

%I A302659 #12 Apr 15 2018 15:03:41
%S A302659 1,4,2,6,8,3,12,7,16,5,24,9,28,10,14,18,20,11,32,15,44,13,36,17,40,19,
%T A302659 52,21,48,22,26,30,34,38,42,46,50,56,23,60,25,64,27,68,29,72,31,76,35,
%U A302659 80,33,84,37,88,41,96,43,92,39,100,45,112,47,116,49,104
%N A302659 Lexicographically earliest sequence of distinct positive terms such that, for any n > 0, A065368(a(n) * a(n+1)) = 0.
%C A302659 For any base b > 1:
%C A302659 - let s_b be the alternate sum of digits in base b,
%C A302659 - in particular s_2 = A065359, s_3 = A065368 and s_10 = A225693,
%C A302659 - we can build an analog of this sequence, say f_b, by considering s_b instead of A065368,
%C A302659 - f_b is well defined: for any k > 0, s_b(k * (1 + b^(2*i + 1))) = 0 whenever k < b^(2*i + 1), hence we can always extend the sequence,
%C A302659 - f_b is conjectured to be a permutation of the natural numbers,
%C A302659 - the scatterplot of f_b shows lines that seem related to the value of s_b(f_b) mod b+1.
%C A302659 See A302544 for a similar sequence.
%H A302659 Rémy Sigrist, <a href="/A302659/b302659.txt">Table of n, a(n) for n = 1..10000</a>
%H A302659 Rémy Sigrist, <a href="/A302659/a302659.png">Colored scatterplot of the first 10000 terms</a>
%H A302659 Rémy Sigrist, <a href="/A302659/a302659_1.png">Scatterplot of the first 10000 of f_2</a>
%H A302659 Rémy Sigrist, <a href="/A302659/a302659_2.png">Scatterplot of the first 10000 of f_5</a>
%H A302659 Rémy Sigrist, <a href="/A302659/a302659.gp.txt">PARI program for A302659</a>
%H A302659 <a href="/index/Di#digital">Index entries for sequences related to digital root, sum, etc.</a>
%e A302659 The first terms, alongside the ternary representation of a(n) * a(n+1), are:
%e A302659   n  a(n)    tern(a(n) * a(n+1))
%e A302659   -- ----    -------------------
%e A302659    1    1        11
%e A302659    2    4        22
%e A302659    3    2       110
%e A302659    4    6      1210
%e A302659    5    8       220
%e A302659    6    3      1100
%e A302659    7   12     10010
%e A302659    8    7     11011
%e A302659    9   16      2222
%e A302659   10    5     11110
%e A302659   11   24     22000
%e A302659   12    9    100100
%o A302659 (PARI) See Links section.
%Y A302659 Cf. A065359, A065368, A225693, A302544.
%K A302659 nonn,base
%O A302659 1,2
%A A302659 _Rémy Sigrist_, Apr 11 2018