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A226723 Positions of the numbers 3^n, for n >= 1, in the joint ranking of all the numbers 2^h, 3^k, 5^k, for h >= 0, k >= 1.

%I A226723 #16 Nov 19 2022 08:57:01
%S A226723 3,7,10,13,16,20,23,26,30,32,36,40,42,46,49,52,55,59,62,65,69,72,75,
%T A226723 79,82,85,88,92,94,98,102,104,108,111,114,118,121,124,127,131,133,137,
%U A226723 141,144,147,150,154,157,160,164,166,170,174,176,180,183,186,189
%N A226723 Positions of the numbers 3^n, for n >= 1, in the joint ranking of all the numbers 2^h, 3^k, 5^k, for h >= 0, k >= 1.
%H A226723 Clark Kimberling, <a href="/A226723/b226723.txt">Table of n, a(n) for n = 1..2000</a>
%F A226723 a(n) = 1 + n + floor(n*log_2(3)) + floor(n*log_5(3)).
%e A226723 The joint ranking of the powers of 2, 3, 5 begins like this: 1, 2, 3, 4, 5, 8, 9, 16, 25, 27, 32, 64, 81, 125, 128, 243, 256, 512.  The numbers 3^n for n >= 1 are in positions 3, 7, 10, 13, 16.
%t A226723 z = 120; b = 2; c = 3; d = 5; f[x_]:=Floor[x];
%t A226723 Table[1 + n + f[n*Log[c, b]] + f[n*Log[d, b]], {n, 0, z}]  (* A226722 *)
%t A226723 Table[1 + n + f[n*Log[b, c]] + f[n*Log[d, c]], {n, 1, z}]  (* A226723 *)
%t A226723 Table[1 + n + f[n*Log[b, d]] + f[n*Log[c, d]], {n, 1, z}]  (* A226724 *)
%Y A226723 Cf. A123384, A226720, A226722, A226724, A306044.
%K A226723 nonn,easy
%O A226723 1,1
%A A226723 _Clark Kimberling_, Jun 16 2013