

A078607


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


9



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
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OFFSET

0,2


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


LINKS



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.
a(n) = floor(1/(11/2^(1/n))).
a(n) = floor(1/(1x)) 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


EXAMPLE

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/(11/sqrt(2))). [From former A230748.]


MATHEMATICA

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


PROG

(PARI) for (n=2, 50, x=2; while (2*x^n<=((x+1)^n), x++); print1(x", "))
(PARI) apply( A078607(n)=ceil(1/if(n>1, sqrtn(2, n)1, !n+n/2)), [0..80]) \\ M. F. Hasler, May 08 2020


CROSSREFS

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


KEYWORD

nonn


AUTHOR



EXTENSIONS

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


STATUS

approved



