A257839 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.

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, 9, 10, 10, 10, 10, 11, 11, 11, 11, 12, 12, 12, 12, 13, 14, 13, 13, 14, 14, 14, 14, 15, 15, 15, 15, 16, 16, 16, 16, 17, 17, 17, 17, 18, 18, 18, 18, 19, 20, 19, 19, 20, 20, 20, 20, 21

Offset: 1

Author

M. F. Hasler, May 16 2015

Keywords

• nonn

Comments

Otherwise said, x-value of the lexicographically first solution (x,y,z) to the given equation.

See A073101 for more details about these sequences related to the Erdős-Straus conjecture.

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.

Links

• M. F. Hasler, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A073101, A075245, A075246, A075247.

Cf. A257840, A257841.

Programs

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

Formula

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