A300654 a(n) is the greatest k such that, for i = 1..k, the binary representation of i appears as a substring in the binary representation of 1/n (ignoring the radix point and adding trailing zeros if necessary in case of a terminating expansion).

2, 2, 2, 2, 4, 2, 2, 2, 4, 4, 8, 2, 9, 2, 2, 2, 4, 4, 16, 4, 4, 8, 6, 2, 8, 9, 11, 2, 20, 2, 2, 2, 4, 4, 8, 4, 32, 16, 6, 4, 4, 4, 8, 8, 6, 6, 12, 2, 12, 8, 2, 9, 33, 11, 10, 2, 8, 20, 37, 2, 41, 2, 2, 2, 4, 4, 64, 4, 14, 8, 14, 4, 4, 32, 11, 16, 17, 6, 22, 4

Offset: 1

Author

Rémy Sigrist, Mar 10 2018

Keywords

• nonn
• base

Comments

Equivalently, a(n) is the greatest k such that A300653(n, k) = k.

This sequence has similarities with A144016: here we consider the binary expansion of 1/n, there the binary expansion of n.

Examples

			For n = 19:
- the binary expansion of 1/19 is 0.0000(110101111001010000) (with repeating digits in parentheses),
- the first occurrence of the binary representation of k for k = 1..16 is:
  k   bin(k)  bin(1/19) with bin(k) in parentheses
  --  ------  ------------------------------------
   1      1   0.0000(1)101...
   2     10   0.00001(10)101...
   3     11   0.0000(11)010...
   4    100   0.000011010111(100)101...
   5    101   0.00001(101)011...
   6    110   0.0000(110)101...
   7    111   0.000011010(111)100...
   8   1000   0.00001101011110010(1000)011...
   9   1001   0.000011010111(1001)010...
  10   1010   0.00001(1010)111...
  11   1011   0.0000110(1011)110...
  12   1100   0.00001101011(1100)101...
  13   1101   0.0000(1101)011...
  14   1110   0.0000110101(1110)010...
  15   1111   0.000011010(1111)001...
  16  10000   0.00001101011110010(10000)110...
- the binary representation of 17 (10001) is missing,
- hence a(19) = 16.
		

Links

• Rémy Sigrist, PARI program for A300654

Crossrefs

Cf. A144016, A300653, A300630.

Programs

PARI

See Links section.

Formula

a(2*n) = a(n).

a(n) = 2 iff n belongs to A300630.