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

2, 4, 10, 27, 80, 230, 644, 1780, 4879, 13315, 36261, 98650, 268260, 729326, 1982655, 5389579, 14650584, 39824632, 108254817, 294267376, 799901968, 2174359323, 5910521810, 16066464445, 43673178798, 118716008808, 322703570021, 877199250941

Offset: 1

Author

Hugo Pfoertner, Aug 21 2017

Keywords

• nonn

Examples

			Let s(x) = x*x^(1/x); r(x) = round(s(x));
a(1) = 2:
  s(1) = 1,
  s(2) = 2.82842712474619...;
  r(1) = 1,
  r(2) = 3,
  r(2) - r(1) = 2;
a(2) = 4:
  s(3) = 4.32674871...,
  s(4) = 5.6568542...;
  r(3) = 4,
  r(4) = 6,
  r(4) - r(3) = 2;
...
a(19) = 108254817:
  s(108254816) = 108254834.49999999422...,
  s(108254817) = 108254835.50000000346...;
  r(108254816) = 108254834,
  r(108254817) = 108254836,
  r(108254817) - r(108254816) = 2.
		

Links

• Hugo Pfoertner, Table of n, a(n) for n = 1..500

Crossrefs

Cf. A000227, A291211, A291212.

Programs

Mathematica

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 *)

Formula

Lim_{n->infinity} a(n)/a(n-1) = e.

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

No more exceptions found through n = 30000. - Hugo Pfoertner, Aug 25 2017