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A291210 Numbers k such that round(k*k^(1/k)) - round((k-1)*(k-1)^(1/(k-1))) > 1.

%I A291210 #35 Aug 27 2017 10:55:51
%S A291210 2,4,10,27,80,230,644,1780,4879,13315,36261,98650,268260,729326,
%T A291210 1982655,5389579,14650584,39824632,108254817,294267376,799901968,
%U A291210 2174359323,5910521810,16066464445,43673178798,118716008808,322703570021,877199250941
%N A291210 Numbers k such that round(k*k^(1/k)) - round((k-1)*(k-1)^(1/(k-1))) > 1.
%H A291210 Hugo Pfoertner, <a href="/A291210/b291210.txt">Table of n, a(n) for n = 1..500</a>
%F A291210 Lim_{n->infinity} a(n)/a(n-1) = e.
%F A291210 It appears that, for most values of n, a(n) = floor(e^(n-1/2) + 7/8) - binomial(n,2). An exception occurs at n = 7; are there more? - _Jon E. Schoenfield_, Aug 22 2017
%F A291210 No more exceptions found through n = 30000. - _Hugo Pfoertner_, Aug 25 2017
%e A291210 Let s(x) = x*x^(1/x); r(x) = round(s(x));
%e A291210 a(1) = 2:
%e A291210   s(1) = 1,
%e A291210   s(2) = 2.82842712474619...;
%e A291210   r(1) = 1,
%e A291210   r(2) = 3,
%e A291210   r(2) - r(1) = 2;
%e A291210 a(2) = 4:
%e A291210   s(3) = 4.32674871...,
%e A291210   s(4) = 5.6568542...;
%e A291210   r(3) = 4,
%e A291210   r(4) = 6,
%e A291210   r(4) - r(3) = 2;
%e A291210 ...
%e A291210 a(19) = 108254817:
%e A291210   s(108254816) = 108254834.49999999422...,
%e A291210   s(108254817) = 108254835.50000000346...;
%e A291210   r(108254816) = 108254834,
%e A291210   r(108254817) = 108254836,
%e A291210   r(108254817) - r(108254816) = 2.
%t A291210 f[n_] := Round[n*n^(1/n)]; g[k_] := f[k] > 1 + f[k-1]; A = Select[Range[2, 5000], g]; Do[AppendTo[A, SelectFirst[Floor[E Last@ A] + Range[1000], g]], {n, 19}]; A (* _Giovanni Resta_, Aug 21 2017 *)
%Y A291210 Cf. A000227, A291211, A291212.
%K A291210 nonn
%O A291210 1,1
%A A291210 _Hugo Pfoertner_, Aug 21 2017