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

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, 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, 3, 73, 9, 1199, 91, 21, 11, 1387, 1, 1, 33, 983, 1, 1, 39, 19, 9

Offset: 0

Author

Rémy Sigrist, Oct 27 2024

Keywords

• nonn
• base
• easy

Comments

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.

Examples

			The first terms, alongside the binary expansion of a(n)*n, are:
  n   a(n)  bin(a(n)*n)
  --  ----  -----------
   0     1            0
   1     1            1
   2     1           10
   3     7        10101
   4     1          100
   5     1          101
   6     7       101010
   7     3        10101
   8     1         1000
   9     9      1010001
  10     1         1010
  11    31    101010101
  12     7      1010100
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..10000
• Rémy Sigrist, C++ program

Crossrefs

See A300867 for a similar sequence.

Cf. A032937, A126684, A195156, A377413.

Programs

Mathematica

(* 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 *)

Formula

a(n) >= A300867(n).

a(n) = 1 iff n belongs to A126684.