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A206812 Position of 2^n in joint ranking of {2^i}, {3^j}, {5^k}.

%I A206812 #18 Nov 18 2022 09:23:57
%S A206812 1,3,5,7,10,11,14,16,17,20,21,24,26,28,30,32,34,36,38,40,43,44,46,49,
%T A206812 50,53,55,57,59,60,63,65,67,69,72,73,75,77,79,82,83,86,88,89,92,94,96,
%U A206812 98,100,102,104,106,108,111,112,115,116,118,121,122,125,127,129
%N A206812 Position of 2^n in joint ranking of {2^i}, {3^j}, {5^k}.
%C A206812 The exponents i,j,k range through the set N of positive integers, so that the position sequences (this sequence for 2^n, A206813 for 3^n, A206814 for 5^n) partition N.
%F A206812 a(n) = n + [n*log_3(2)] + [n*log_5(2)],
%F A206812 A206813(n) = n + [n*log_2(3)] + [n*log_5(3)],
%F A206812 A206814(n) = n + [n*log_2(5)] + [n*log_3(5)],
%F A206812 where []=floor.
%e A206812 The joint ranking begins with 2,3,4,5,8,9,16,25,27,32,64,81,125,128,243 so that
%e A206812 this sequence = (1,3,5,7,10,11,...)
%e A206812 A206813 = (2,6,9,12,15,...)
%e A206812 A206814 = (4,8,13,18,23,...)
%t A206812 f[1, n_] := 2^n; f[2, n_] := 3^n;
%t A206812 f[3, n_] := 5^n; z = 1000;
%t A206812 d[n_, b_, c_] := Floor[n*Log[b, c]];
%t A206812 t[k_] := Table[f[k, n], {n, 1, z}];
%t A206812 t = Sort[Union[t[1], t[2], t[3]]];
%t A206812 p[k_, n_] := Position[t, f[k, n]];
%t A206812 Flatten[Table[p[1, n], {n, 1, z/8}]] (* A206812 *)
%t A206812 Table[n + d[n, 3, 2] + d[n, 5, 2],
%t A206812   {n, 1, 50}]                        (* A206812 *)
%t A206812 Flatten[Table[p[2, n], {n, 1, z/8}]] (* A206813 *)
%t A206812 Table[n + d[n, 2, 3] + d[n, 5, 3],
%t A206812   {n, 1, 50}]                        (* A206813 *)
%t A206812 Flatten[Table[p[3, n], {n, 1, z/8}]] (* A206814 *)
%t A206812 Table[n + d[n, 2, 5] + d[n, 3, 5],
%t A206812   {n, 1, 50}]                        (* A206814 *)
%o A206812 (PARI) a(n) = n + logint(2^n,3) + logint(2^n,5) \\ _Ruud H.G. van Tol_, Nov 16 2022
%Y A206812 Cf. A206805, A206813, A206814.
%K A206812 nonn
%O A206812 1,2
%A A206812 _Clark Kimberling_, Feb 17 2012