A078607 Least positive integer x such that 2*x^n > (x+1)^n.

1, 2, 3, 4, 6, 7, 9, 10, 12, 13, 14, 16, 17, 19, 20, 22, 23, 25, 26, 27, 29, 30, 32, 33, 35, 36, 38, 39, 40, 42, 43, 45, 46, 48, 49, 50, 52, 53, 55, 56, 58, 59, 61, 62, 63, 65, 66, 68, 69, 71, 72, 74, 75, 76, 78, 79, 81, 82, 84, 85, 87, 88, 89, 91, 92, 94, 95, 97, 98, 100, 101, 102

Offset: 0

Author

Jon Perry, Dec 09 2002

Keywords

• nonn

Comments

Also, integer for which E(s) = s^n - Sum_{0 < k < s} k^n is maximal. It appears that a(n) + 2 is the least integer for which E(s) < 0. - M. F. Hasler, May 08 2020

Examples

			a(2) = 3 as 2^2 = 4, 3^2 = 9 and 4^2 = 16.
For n = 777451915729368, a(n) = 1121626023352384 = ceiling(n log 2), where n*log(2) = 1121626023352383.5 - 2.13*10^-17 and 2*floor(n log 2)^n / floor(1 + n log 2)^n = 1 - 3.2*10^-32. - _M. F. Hasler_, Nov 02 2013
a(2) is given by floor(1/(1-1/sqrt(2))). [From former A230748.]
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..20000

Crossrefs

Cf. A224996 (the largest integer x that satisfies 2*x^n <= (x+1)^n).

Cf. A050499, A050500.

Cf. A078608, A078609. Equals A110882(n)-1 for n > 0.

Cf. A332097 (maximum of E(s), cf comments), also related to this: A332101 (least k such that k^n <= sum of all smaller n-th powers), A030052 (least k such that k^n = sum of distinct n-th powers), A332065 (all k such that k^n is a sum of distinct n-th powers).

Programs

Mathematica

Table[SelectFirst[Range@ 120, 2 #^n > (# + 1)^n &], {n, 0, 71}] (* Michael De Vlieger, May 01 2016, Version 10 *)

PARI

for (n=2,50, x=2; while (2*x^n<=((x+1)^n),x++); print1(x","))

PARI

a(n)=1\(1-1/2^(1/n)) \\ Charles R Greathouse IV, Oct 31 2013

PARI

apply( A078607(n)=ceil(1/if(n>1,sqrtn(2,n)-1,!n+n/2)), [0..80]) \\ M. F. Hasler, May 08 2020

Formula

a(n) = ceiling(1/(2^(1/n)-1)) for n > 1. (For n = 1 resp. 0 this gives the integer 1 resp. infinity as argument of ceiling.) [Edited by M. F. Hasler, May 08 2020]

For most n, a(n) is the nearest integer to n/log(2), but there are exceptions, including n=777451915729368.

Following formulae merged in from former A230748, M. F. Hasler, May 14 2020:

a(n) = floor(1/(1-1/2^(1/n))).

a(n) = n/log(2) + O(1). - Charles R Greathouse IV, Oct 31 2013

a(n) = floor(1/(1-x)) with x^n = 1/2: f(n) = 1/(2^(1/n)-1) is never an integer for n > 1, whence floor(f(n)+1) = ceiling(f(n)) = a(n). - M. F. Hasler, Nov 02 2013, and Gabriel Conant, May 01 2016

Extensions

Edited by Dean Hickerson, Dec 17 2002

Initial terms a(0) = 1 and a(1) = 2 added by M. F. Hasler, Nov 02 2013