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A377412 a(n) is the least k > 0 such that k*n belongs to A126684.

%I A377412 #10 Oct 30 2024 13:55:41
%S A377412 1,1,1,7,1,1,7,3,1,9,1,31,7,5,3,91,1,1,9,55,1,1,31,3,7,13,5,3,3,9,91,
%T A377412 11,1,33,1,39,9,113,55,7,1,25,1,127,31,121,3,443,7,21,13,87,5,97,3,19,
%U A377412 3,73,9,1199,91,21,11,1387,1,1,33,983,1,1,39,19,9
%N A377412 a(n) is the least k > 0 such that k*n belongs to A126684.
%C A377412 This sequence is well defined: for any positive integer n, according to the pigeonhole principle, A195156(i) mod n = A195156(j) mod n for some distinct i and j, hence n divides b = abs(A195156(i) - A195156(j)), and as b belongs to A126684, a(n) <= b/n.
%H A377412 Rémy Sigrist, <a href="/A377412/b377412.txt">Table of n, a(n) for n = 0..10000</a>
%H A377412 Rémy Sigrist, <a href="/A377412/a377412.txt">C++ program</a>
%F A377412 a(n) >= A300867(n).
%F A377412 a(n) = 1 iff n belongs to A126684.
%e A377412 The first terms, alongside the binary expansion of a(n)*n, are:
%e A377412   n   a(n)  bin(a(n)*n)
%e A377412   --  ----  -----------
%e A377412    0     1            0
%e A377412    1     1            1
%e A377412    2     1           10
%e A377412    3     7        10101
%e A377412    4     1          100
%e A377412    5     1          101
%e A377412    6     7       101010
%e A377412    7     3        10101
%e A377412    8     1         1000
%e A377412    9     9      1010001
%e A377412   10     1         1010
%e A377412   11    31    101010101
%e A377412   12     7      1010100
%t A377412 (* Increase nmax for n>92 in A377412 *) nmax = 1000; b[n_] := FromDigits[IntegerDigits[n, 2], 4];A126684=Union[A000695 = b /@ Range[0, nmax], 2 A000695][[1 ;; nmax+1]] ;A377412={};Do[k=0;Until[MemberQ[A126684,k*n],k++]; AppendTo[A377412,k],{n,0,72}];A377412 (* _James C. McMahon_, Oct 29 2024 *)
%o A377412 (C++) // See Links section.
%Y A377412 See A300867 for a similar sequence.
%Y A377412 Cf. A032937, A126684, A195156, A377413.
%K A377412 nonn,base,easy
%O A377412 0,4
%A A377412 _Rémy Sigrist_, Oct 27 2024