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A300867 a(n) is the least positive k such that k * n is a Fibbinary number (A003714).

%I A300867 #15 Oct 28 2024 16:24:12
%S A300867 1,1,1,3,1,1,3,3,1,1,1,3,3,5,3,11,1,1,1,7,1,1,3,3,3,13,5,3,3,5,11,11,
%T A300867 1,1,1,39,1,1,7,7,1,1,1,3,3,13,3,7,3,21,13,23,5,5,3,3,3,9,5,11,11,9,
%U A300867 11,43,1,1,1,35,1,1,39,15,1,1,1,31,7,57,7,7,1
%N A300867 a(n) is the least positive k such that k * n is a Fibbinary number (A003714).
%C A300867 This sequence is well defined: for any positive n, according to the pigeonhole principle, A195156(i) mod n = A195156(j) mod n for some distinct i and j, hence n divides f = abs(A195156(i) - A195156(j)), and as f is a Fibbinary number, a(n) <= f/n.
%C A300867 All terms are odd.
%H A300867 Rémy Sigrist, <a href="/A300867/b300867.txt">Table of n, a(n) for n = 0..10000</a>
%H A300867 Rémy Sigrist, <a href="/A300867/a300867.png">Colored logarithmic scatterplot of the first 1000000 terms</a> (where the color is function of A070939(n * a(n)))
%F A300867 a(n) = A300889(n) / n for any n > 0.
%F A300867 a(2*n) = a(n).
%F A300867 a(n) = 1 iff n belongs to A003714.
%e A300867 The first terms, alongside the binary representation of n * a(n), are:
%e A300867   n  a(n)   bin(n * a(n))
%e A300867   -- ----   -------------
%e A300867    0    1           0
%e A300867    1    1           1
%e A300867    2    1          10
%e A300867    3    3        1001
%e A300867    4    1         100
%e A300867    5    1         101
%e A300867    6    3       10010
%e A300867    7    3       10101
%e A300867    8    1        1000
%e A300867    9    1        1001
%e A300867   10    1        1010
%e A300867   11    3      100001
%e A300867   12    3      100100
%e A300867   13    5     1000001
%e A300867   14    3      101010
%e A300867   15   11    10100101
%e A300867   16    1       10000
%e A300867   17    1       10001
%e A300867   18    1       10010
%e A300867   19    7    10000101
%e A300867   20    1       10100
%o A300867 (PARI) a(n) = forstep (k=1, oo, 2, if (bitand(k*n, 2*k*n)==0, return (k)))
%Y A300867 Cf. A003714, A070939, A195156, A300889.
%K A300867 nonn,look,base
%O A300867 0,4
%A A300867 _Rémy Sigrist_, Mar 14 2018