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

Author

Clark Kimberling, Jun 16 2013

Keywords

• nonn
• easy

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).

Examples

			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.
		

Links

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

Crossrefs

Cf. A123384, A226721.

Programs

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 *)