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A226722 Positions of the numbers 2^n, for n >=0, in the joint ranking of all the numbers 2^h, 3^k, 5^k, for h >= 0, k >= 1.

%I A226722 #18 Nov 03 2024 02:24:14
%S A226722 1,2,4,6,8,11,12,15,17,18,21,22,25,27,29,31,33,35,37,39,41,44,45,47,
%T A226722 50,51,54,56,58,60,61,64,66,68,70,73,74,76,78,80,83,84,87,89,90,93,95,
%U A226722 97,99,101,103,105,107,109,112,113,116,117,119,122,123,126
%N A226722 Positions of the numbers 2^n, for n >=0, in the joint ranking of all the numbers 2^h, 3^k, 5^k, for h >= 0, k >= 1.
%H A226722 Clark Kimberling, <a href="/A226722/b226722.txt">Table of n, a(n) for n = 1..2000</a>
%F A226722 a(n) = n + floor((n-1)*log_3(2)) + floor((n-1)*log_5(2)). [corrected by _Jason Yuen_, Nov 02 2024]
%e A226722 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 2^n for n >= 0 are in positions 1, 2, 4, 6, 8, 11, 12, 15, 17, 18.
%t A226722 z = 120; b = 2; c = 3; d = 5; f[x_]:=Floor[x];
%t A226722 Table[n + f[(n-1)*Log[c, b]] + f[(n-1)*Log[d, b]], {n, 1, z}]  (* this sequence *)
%t A226722 Table[1 + n + f[n*Log[b, c]] + f[n*Log[d, c]], {n, 1, z}]  (* A226723 *)
%t A226722 Table[1 + n + f[n*Log[b, d]] + f[n*Log[c, d]], {n, 1, z}]  (* A226724 *)
%Y A226722 Cf. A123384, A226720, A226723, A226724, A306044.
%K A226722 nonn,easy
%O A226722 1,2
%A A226722 _Clark Kimberling_, Jun 16 2013