A326538 a(n) is the numerator of the image of 1/n by the Cantor staircase function.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 3, 3, 3, 3, 23, 1, 5, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 15, 1, 7, 7

Offset: 1

Author

Rémy Sigrist, Jul 12 2019

Keywords

• nonn
• base
• frac

Comments

The Cantor staircase function, say c, maps rational numbers in the interval [0..1] to rational numbers in the interval [0..1], hence this sequence is well defined.

For any n > 0, the binary expansion of c(1/n) is terminating (and A326539(n) is a power of 2) iff the ternary expansion of 1/n is terminating or contains a digit 1.

Examples

			The first terms, alongside c(1/n) and the ternary and binary representation of 1/n and c(1/n), respectively, with periodic part in parentheses, are:
  n   a(n)  c(1/n)  ter(1/n)                bin(c(1/n))
  --  ----  ------  ----------------------  -----------
   1     1       1  1.(0)                   1.(0)
   2     1     1/2  0.(1)                   0.1(0)
   3     1     1/2  0.1(0)                  0.1(0)
   4     1     1/3  0.(02)                  0.(01)
   5     1     1/4  0.(0121)                0.01(0)
   6     1     1/4  0.0(1)                  0.01(0)
   7     1     1/4  0.(010212)              0.01(0)
   8     1     1/4  0.(01)                  0.01(0)
   9     1     1/4  0.01(0)                 0.01(0)
  10     1     1/5  0.(0022)                0.(0011)
  11     3    3/16  0.(00211)               0.0011(0)
  12     1     1/6  0.0(02)                 0.0(01)
  13     1     1/7  0.(002)                 0.(001)
  14     1     1/8  0.(001221)              0.001(0)
  15     1     1/8  0.0(0121)               0.001(0)
  16     1     1/8  0.(0012)                0.001(0)
  17     1     1/8  0.(0011202122110201)    0.001(0)
  18     1     1/8  0.00(1)                 0.001(0)
  19     1     1/8  0.(001102100221120122)  0.001(0)
  20     1     1/8  0.(0011)                0.001(0)
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..6561
• Rémy Sigrist, PARI program for A326538
• Wikipedia, Cantor function

Crossrefs

See A326539 for the corresponding denominators.

Cf. A061392.

Programs

PARI

See Links section.