1,1

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).

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

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.

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

Floor[1 + Range[1, 100]*(1 + Log[c, b])] (* A226721 *)

Cf. A123384, A226721.

Sequence in context: A026351 A184656 A286989 * A047212 A358845 A121347

Adjacent sequences: A226717 A226718 A226719 * A226721 A226722 A226723

nonn,easy

Clark Kimberling, Jun 16 2013

approved