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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
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OFFSET
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1,1
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COMMENTS
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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).
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LINKS
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EXAMPLE
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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.
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MATHEMATICA
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b = 2; c=3; Floor[1 + Range[0, 100]*(1 + Log[b, c])] (* A123384 *)
Floor[1 + Range[1, 100]*(1 + Log[c, b])] (* A226721 *)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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