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%I #30 May 15 2020 13:00:42
%S 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,
%T 36,38,39,40,42,43,45,46,48,49,50,52,53,55,56,58,59,61,62,63,65,66,68,
%U 69,71,72,74,75,76,78,79,81,82,84,85,87,88,89,91,92,94,95,97,98,100,101,102
%N Least positive integer x such that 2*x^n > (x+1)^n.
%C 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
%F 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]
%F For most n, a(n) is the nearest integer to n/log(2), but there are exceptions, including n=777451915729368.
%F Following formulae merged in from former A230748, _M. F. Hasler_, May 14 2020:
%F a(n) = floor(1/(1-1/2^(1/n))).
%F a(n) = n/log(2) + O(1). - _Charles R Greathouse IV_, Oct 31 2013
%F 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
%e a(2) = 3 as 2^2 = 4, 3^2 = 9 and 4^2 = 16.
%e 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
%e a(2) is given by floor(1/(1-1/sqrt(2))). [From former A230748.]
%t Table[SelectFirst[Range@ 120, 2 #^n > (# + 1)^n &], {n, 0, 71}] (* _Michael De Vlieger_, May 01 2016, Version 10 *)
%o (PARI) for (n=2,50, x=2; while (2*x^n<=((x+1)^n),x++); print1(x","))
%o (PARI) a(n)=1\(1-1/2^(1/n)) \\ _Charles R Greathouse IV_, Oct 31 2013
%o (PARI) apply( A078607(n)=ceil(1/if(n>1,sqrtn(2,n)-1,!n+n/2)), [0..80]) \\ _M. F. Hasler_, May 08 2020
%Y Cf. A050499, A050500.
%Y Cf. A078608, A078609. Equals A110882(n)-1 for n > 0.
%Y 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).
%K nonn
%O 0,2
%A _Jon Perry_, Dec 09 2002
%E Edited by _Dean Hickerson_, Dec 17 2002
%E Initial terms a(0) = 1 and a(1) = 2 added by _M. F. Hasler_, Nov 02 2013
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