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A347069 Rectangular array (T(n,k)), by downward antidiagonals: T(n,k) = position of k in the ordering of {h*e^m, h >= 1, 0 <= m <= n}.

%I A347069 #7 Nov 20 2021 21:24:53
%S A347069 1,2,1,4,2,1,5,4,2,1,6,5,4,2,1,8,6,5,4,2,1,9,8,6,5,4,2,1,10,9,8,6,5,4,
%T A347069 2,1,12,11,9,8,6,5,4,2,1,13,13,11,9,8,6,5,4,2,1,15,14,13,11,9,8,6,5,4,
%U A347069 2,1,16,16,14,13,11,9,8,6,5,4,2,1,17,17
%N A347069 Rectangular array (T(n,k)), by downward antidiagonals: T(n,k) = position of k in the ordering of {h*e^m, h >= 1, 0 <= m <= n}.
%C A347069 No two rows are identical.
%e A347069 m = 0 gives 1, 2, 3, 4, 5, 6, ...
%e A347069 m = 1 gives e, 2e, 3e, 4e, 5e, ...
%e A347069 Row 1 of the array tells the positions of the positive integers when the numbers for m=0 and m=1 are jointly ranked.  Using decimal approximations, the numbers, jointly ranked, are 1, 2, 2.718, 3, 4, 5, 6.436, 6, 7, 8, 8.154, 9, 10, 10.873, 11, ...
%e A347069 Corner:
%e A347069   1, 2, 4, 5, 6, 8, 9, 10, 12, 13, 15, 16, 17
%e A347069   1, 2, 4, 5, 6, 8, 9, 11, 13, 14, 16, 17, 18
%e A347069   1, 2, 4, 5, 6, 8, 9, 11, 13, 14, 16, 17, 18
%e A347069   1, 2, 4, 5, 6, 8, 9, 11, 13, 14, 16, 17, 18
%e A347069   1, 2, 4, 5, 6, 8, 9, 11, 13, 14, 16, 17, 18
%e A347069   1, 2, 4, 5, 6, 8, 9, 11, 13, 14, 16, 17, 18
%e A347069   1, 2, 4, 5, 6, 8, 9, 11, 13, 14, 16, 17, 18
%t A347069 z = 100; r = N[E];
%t A347069 s[m_] := Range[z] r^m; t[0] = s[0];
%t A347069 t[n_] := Sort[Union[s[n], t[n - 1]]]
%t A347069 row[n_] := Flatten[Table[Position[t[n], N[k]], {k, 1, z}]]
%t A347069 TableForm[Table[row[n], {n, 1, 10}]] (* A347069, array *)
%t A347069 w[n_, k_] := row[n][[k]];
%t A347069 Table[w[n - k + 1, k], {n, 12}, {k, n, 1, -1}] // Flatten (* A347069, sequence *)
%Y A347069 Cf. A347065, A347066, A347067, A347068.
%K A347069 nonn,tabl
%O A347069 1,2
%A A347069 _Clark Kimberling_, Sep 02 2021