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A226724 Positions of the numbers 5^n, for n >= 1, in the joint ranking of all the numbers 2^h, 3^k, 5^k, for h >= 0, k >= 1.

%I A226724 #15 Nov 02 2024 18:00:46
%S A226724 5,9,14,19,24,28,34,38,43,48,53,57,63,67,71,77,81,86,91,96,100,106,
%T A226724 110,115,120,125,129,135,139,143,148,153,158,162,168,172,177,182,187,
%U A226724 191,197,201,205,211,215,220,225,230,234,240,244,249,254,259,263,269
%N A226724 Positions of the numbers 5^n, for n >= 1, in the joint ranking of all the numbers 2^h, 3^k, 5^k, for h >= 0, k >= 1.
%H A226724 Clark Kimberling, <a href="/A226724/b226724.txt">Table of n, a(n) for n = 1..2000</a>
%F A226724 a(n) = 1 + n + floor(n*log_2(5)) + floor(n*log_3(5)).
%e A226724 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 5^n for n >= 0 are in positions 5, 9, 14.
%t A226724 z = 120; b = 2; c = 3; d = 5; f[x_]:=Floor[x];
%t A226724 Table[1 + n + f[n*Log[c, b]] + f[n*Log[d, b]], {n, 0, z}]  (* A226722 *)
%t A226724 Table[1 + n + f[n*Log[b, c]] + f[n*Log[d, c]], {n, 1, z}]  (* A226723 *)
%t A226724 Table[1 + n + f[n*Log[b, d]] + f[n*Log[c, d]], {n, 1, z}]  (* A226724 *)
%Y A226724 Cf. A123384, A226720, A226722, A226723, A306044.
%K A226724 nonn,easy
%O A226724 1,1
%A A226724 _Clark Kimberling_, Jun 16 2013