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%I #64 Aug 31 2023 13:37:16
%S 3,5,6,9,10,13,14,17,18,19,21,26,29,30,38,41,51,53,54,57,66,67,69,73,
%T 74,77,83,86,94,101,102,105,106,110,113,117,122,126,129,130,133,142,
%U 145,147,149,154,158,161,162,166,174,177,178,181,186,195,197,201,206
%N Integers expressible as (x^3 + y^3 + z^3)/(x*y*z) with positive integers x, y and z. (Alternatively, the integers expressible as x/y + y/z + z/x with positive integers x, y and z.)
%C It is easy to see that the numbers of the form m^2 + 5 are included in this sequence for m = 0, 1, 2, .... It is known that no number of any of the forms below appears in the sequence: (a) multiples of 4, (b) numbers congruent to 7 mod 8, (c) numbers of the form 2^m*k + 3 with odd numbers m >= 3 and k >= 1. It is also known, on the other hand, that there are infinitely many numbers congruent to 1, 2, 3, 5, or 6 mod 8 included in the sequence.
%C There are other parametric representations (like m^2 + 5) with positive a, b and c: (see example below for generating expressions): k = (m^2 + 33)/2: 17, 21, 29, 41, 57, 77, 101, 129, 161, 197, 237, 281, 329, 381, 437, 497, ...; k = m^4 + 8*m^2 + 4*m + 13: ..., 818, 381, 154, 53, 18, 13, 26, 69, 178, 413, 858, ...; k = m^4 - 2*m^3 + 4*m^2 + 3: ..., 978, 451, 174, 51, 10, 3, 6, 19, 66, 195, 478, ... - Erik Dofs (erik.dofs(AT)swipnet.se), Mar 06 2004
%H Jinyuan Wang, <a href="/A072716/b072716.txt">Table of n, a(n) for n = 1..250</a>
%H Andrew Bremner and R. K. Guy, <a href="https://doi.org/10.1017/S0013091500023397">Two more representation problems</a>, Proc. Edinburgh Math. Soc. vol. 40 1997 pp. 1-17.
%H Erik Dofs, <a href="https://doi.org/10.4064/aa-73-3-201-213">Solutions of x^3+y^3+z^3=nxyz</a>, Acta Arith. LXXIII.3 (1995).
%H Erik Dofs and Nguyen Xuan Tho, <a href="https://doi.org/10.1142/S1793042122500075">The equation x_1/x_2 + x_2/x_3 + x_3/x_4 + x_4/x_1 = n</a>, Int'l J. Num. Theory (2021). See also <a href="https://www.researchgate.net/profile/Tho-Nguyen-18/publication/350958036_The_Diophantine_equation_x_1x_2x_2x_3x_3x_4x_4x_1n">on ResearchGate</a>.
%H David J. Rusin, <a href="https://web.archive.org/web/20130619194421/http://www.math.niu.edu/~rusin/research-math/abcn/">For which values of n is a/b + b/c + c/a = n solvable?</a> [Archive Machine link]
%H David J. Rusin, <a href="/A072716/a072716.pdf">For which values of n is a/b + b/c + c/a = n solvable?</a> [Cached copy of html wrapper for paper but in pdf format (so none of the links work)]
%H David J. Rusin, <a href="/A072716/a072716.txt">For which values of n is a/b + b/c + c/a = n solvable?</a> [Cached copy of .txt file]
%e {k, x, y, z} = {(m^2 + 33)/2, (m^4 + 6m^3 + 36m^2 + 98m + 147)/16, (m^4 - 6m^3 + 36m^2 - 98m + 147)/16, (m^2 + 147)/4}/GCD[(m^4 + 6m^3 + 36m^2 + 98m + 147)/16, (m^4 - 6m^3 + 36m^2 - 98m + 147)/16]}
%e {k, x, y, z} = {m^4 + 8m^2 + 4m + 13, m^6 + m^5 + 10m^4 + 11m^3 + 28m^2 + 27m + 13, m^6 - 3m^5 + 12m^4 - 19m^3 + 30m^2 - 21m + 9, 2m^2 - 2m + 38}/ GCD[m^6 + m^5 + 10m^4 + 11m^3 + 28m^2 + 27m + 13, m^6 - 3m^5 + 12m^4 - 19m^3 + 30m^2 - 21m + 9]}
%e {k, x, y, z} = {m^4 - 2m^3 + 4m^2 + 3, m^4 - 3m^3 + 6m^2 - 5m + 3, m^2 - m + 3, m^2 - 3m + 3}/GCD[m^2 - m + 3, m^2 - 3m + 3]}
%e 41 appears in the sequence because we can write 41 = (1^3 + 2^3 + 9^3)/(1*2*9).
%e For n = 142, {x,y,z} = {6587432496387235561093636933115859813174, 53881756527432415186060525094013536917351, 222932371699623861287567763383948430761525}.
%Y Cf. A085705.
%K nonn,nice,hard
%O 1,1
%A Tadaaki Ohno (t-ohno(AT)hyper.ocn.ne.jp), Aug 07 2002
%E More terms from _David J. Rusin_, Jul 26 2003
%E a(52)-a(58) from _Jinyuan Wang_, Jul 27 2021
%E a(59) from _Jinyuan Wang_, Aug 31 2023
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