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A206813 Position of 3^n in joint ranking of {2^i}, {3^j}, {5^k}.

%I A206813 #5 Mar 30 2012 18:58:12
%S A206813 2,6,9,12,15,19,22,25,29,31,35,39,41,45,48,51,54,58,61,64,68,71,74,78,
%T A206813 81,84,87,91,93,97,101,103,107,110,113,117,120,123,126,130,132,136,
%U A206813 140,143,146,149,153,156,159,163,165,169,173,175,179,182,185,188
%N A206813 Position of 3^n in joint ranking of {2^i}, {3^j}, {5^k}.
%C A206813 The exponents i,j,k range through the set N of positive integers, so that the position sequences (A206812 for 2^n, A206813 for 3^n, A206814 for 5^n) partition N.
%F A206813 A205812(n) = n + [n*log(base 3)(2)] + [n*log(base 5)(2)],
%F A206813 A205813(n) = n + [n*log(base 2)(3)] + [n*log(base 5)(3)],
%F A206813 A205814(n) = n + [n*log(base 2)(5)] + [n*log(base 3)(5)],
%F A206813 where []=floor.
%e A206813 The joint ranking begins with 2,3,4,5,8,9,16,25,27,32,64,81,125,128,243,256, so that
%e A206813 A205812=(1,3,5,7,10,11,14,...)
%e A206813 A205813=(2,6,9,12,15,...)
%e A206813 A205814=(4,8,13,18,23,...)
%t A206813 f[1, n_] := 2^n; f[2, n_] := 3^n;
%t A206813 f[3, n_] := 5^n; z = 1000;
%t A206813 d[n_, b_, c_] := Floor[n*Log[b, c]];
%t A206813 t[k_] := Table[f[k, n], {n, 1, z}];
%t A206813 t = Sort[Union[t[1], t[2], t[3]]];
%t A206813 p[k_, n_] := Position[t, f[k, n]];
%t A206813 Flatten[Table[p[1, n], {n, 1, z/8}]] (* A206812 *)
%t A206813 Table[n + d[n, 3, 2] + d[n, 5, 2],
%t A206813   {n, 1, 50}]                        (* A206812 *)
%t A206813 Flatten[Table[p[2, n], {n, 1, z/8}]] (* A206813 *)
%t A206813 Table[n + d[n, 2, 3] + d[n, 5, 3],
%t A206813   {n, 1, 50}]                        (* A206813 *)
%t A206813 Flatten[Table[p[3, n], {n, 1, z/8}]] (* A206814 *)
%t A206813 Table[n + d[n, 2, 5] + d[n, 3, 5],
%t A206813   {n, 1, 50}]                        (* A206814 *)
%Y A206813 Cf. A206805, A206812, A206814.
%K A206813 nonn
%O A206813 1,1
%A A206813 _Clark Kimberling_, Feb 17 2012