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%I #21 Jul 04 2022 04:45:03
%S 0,0,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,6,7,7,7,7,8,8,8,8,9,9,9,
%T 9,10,10,10,10,11,11,11,11,12,12,12,12,13,14,13,13,14,14,14,14,15,15,
%U 15,15,16,16,16,16,17,17,17,17,18,18,18,18,19,20,19,19,20,20,20,20,21
%N Smallest possible x such that 4/n = 1/x + 1/y + 1/z with 0 < x < y < z all integers, or 0 if there is no such solution. Corresponding y and z values are in A257840 and A257841.
%C Otherwise said, x-value of the lexicographically first solution (x,y,z) to the given equation.
%C See A073101 for more details about these sequences related to the Erdős-Straus conjecture.
%C This differs from A075245 starting with a(89)=23 vs A075245(89)=24, respective solutions being 1/23 + 1/690 + 1/61410 = 1/24 + 1/306 + 1/108936 = 4/89.
%H M. F. Hasler, <a href="/A257839/b257839.txt">Table of n, a(n) for n = 1..1000</a>
%F Conjecture: a(n) = floor(n/4) + d with d = 1 for all n > 2 except some n = 24k + 1 (k = 2, 3, 7, 8, 10, 13, 15, 17, 18, 23, 25, 28, 30, 32, 33, 37, 40, 43, ...) where d = 2. - _M. F. Hasler_, Jul 03 2022
%o (PARI) apply( {A257839(n, t)=for(x=n\4+1, 3*n\4, for(y=max(1\t=4/n-1/x, x)+1, ceil(2/t)-1, numerator(t-1/y)==1 && return(x)))}, [1..99]) \\ improved by _M. F. Hasler_, Jul 03 2022
%Y Cf. A073101, A075245, A075246, A075247.
%Y Cf. A257840, A257841.
%K nonn
%O 1,4
%A _M. F. Hasler_, May 16 2015
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