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

0,4

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.

Links

Rémy Sigrist, Table of n, a(n) for n = 0..10000

Rémy Sigrist, C++ program

Formula

a(n) >= A300867(n).

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

Example

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

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 *)

Prog

(C++) // See Links section.

Crossrefs

See A300867 for a similar sequence.

Cf. A032937, A126684, A195156, A377413.

Sequence in context: A019620 A105395 A120437 * A336459 A367764 A174095

Adjacent sequences: A377409 A377410 A377411 * A377413 A377414 A377415

Keyword

nonn,base,easy

Author

Rémy Sigrist, Oct 27 2024

Status

approved