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A291210 Numbers k such that round(k*k^(1/k)) - round((k-1)*(k-1)^(1/(k-1))) > 1. 3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

LINKS

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

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

EXAMPLE

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.

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 *)

CROSSREFS

Cf. A000227, A291211, A291212.

Sequence in context: A127386 A279551 A148106 * A340635 A099950 A121690

Adjacent sequences:  A291207 A291208 A291209 * A291211 A291212 A291213

KEYWORD

nonn

AUTHOR

Hugo Pfoertner, Aug 21 2017

STATUS

approved

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Last modified June 15 00:00 EDT 2021. Contains 345041 sequences. (Running on oeis4.)