A300867 -id:A300867 - cp's OEIS frontend

A300889 a(n) is the least positive multiple of n which is a Fibbinary number (A003714).

1, 2, 9, 4, 5, 18, 21, 8, 9, 10, 33, 36, 65, 42, 165, 16, 17, 18, 133, 20, 21, 66, 69, 72, 325, 130, 81, 84, 145, 330, 341, 32, 33, 34, 1365, 36, 37, 266, 273, 40, 41, 42, 129, 132, 585, 138, 329, 144, 1029, 650, 1173, 260, 265, 162, 165, 168, 513, 290, 649

Offset: 1

Rémy Sigrist, Mar 14 2018

			The first terms, alongside their binary representation, are:
  n  a(n)    binary(a(n))
  -- ----    ------------
   1    1           1
   2    2          10
   3    9        1001
   4    4         100
   5    5         101
   6   18       10010
   7   21       10101
   8    8        1000
   9    9        1001
  10   10        1010
  11   33      100001
  12   36      100100
  13   65     1000001
  14   42      101010
  15  165    10100101
  16   16       10000
  17   17       10001
  18   18       10010
  19  133    10000101
  20   20       10100

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Cf. A003714, A300867.

a(n) = forstep (k=1, oo, 2, if (bitand(k*n, 2*k*n)==0, return (k*n)))

a(n) = n * A300867(n).
a(2*n) = 2*a(n).
a(n) = n iff n belongs to A003714.

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

Original entry on oeis.org

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

Rémy Sigrist, Oct 27 2024

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.

			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

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

See A300867 for a similar sequence.

Cf. A032937, A126684, A195156, A377413.

(* 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 *)

a(n) >= A300867(n).

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

cp's OEIS Frontend

A300889 a(n) is the least positive multiple of n which is a Fibbinary number (A003714).

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