A226720 Complement of A122437.

2, 4, 5, 7, 9, 10, 12, 14, 15, 17, 18, 20, 22, 23, 25, 27, 28, 30, 31, 33, 35, 36, 38, 40, 41, 43, 45, 46, 48, 49, 51, 53, 54, 56, 58, 59, 61, 62, 64, 66, 67, 69, 71, 72, 74, 76, 77, 79, 80, 82, 84, 85, 87, 89, 90, 92, 93, 95, 97, 98, 100, 102, 103, 105, 107

Offset

1,1

Comments

Suppose that b and c are integers satisfying 1 < b < c. Let x = 1 + log_b(c) and y = 1 + log_c(b). Jointly rank all the numbers b^k for k>=0 and c^k for k>=1; then for n >= 0, the position of b^n is 1 + floor(n*y), and for n >=1, the position of c^n is 1+ floor(n*x).

These position sequences are closely related to the Beatty sequences given by floor(n*x) and floor(n*y).

Links

Clark Kimberling, Table of n, a(n) for n = 1..2000

Example

The joint ranking of the powers of 2 and of 3 begins like this: 1, 2, 3, 4, 8, 9, 16, 27, 32, 64. The numbers 2^n for n >= 1 are in positions 2, 4, 5, 7, 9, 10.

Mathematica

b = 2; c=3; Floor[1 + Range[0, 100]*(1 + Log[b, c])]  (* A123384 *)
Floor[1 + Range[1, 100]*(1 + Log[c, b])]  (* A226721 *)

Crossrefs

Cf. A123384, A226721.

Sequence in context: A026351 A184656 A286989 * A047212 A358845 A121347

Adjacent sequences: A226717 A226718 A226719 * A226721 A226722 A226723

Keyword

nonn,easy

Author

Clark Kimberling, Jun 16 2013

Status

approved