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A072716
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.)
2
3, 5, 6, 9, 10, 13, 14, 17, 18, 19, 21, 26, 29, 30, 38, 41, 51, 53, 54, 57, 66, 67, 69, 73, 74, 77, 83, 86, 94, 101, 102, 105, 106, 110, 113, 117, 122, 126, 129, 130, 133, 142, 145, 147, 149, 154, 158, 161, 162, 166, 174, 177, 178, 181, 186, 195, 197, 201, 206
OFFSET
1,1
COMMENTS
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.
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
LINKS
Andrew Bremner and R. K. Guy, Two more representation problems, Proc. Edinburgh Math. Soc. vol. 40 1997 pp. 1-17.
Erik Dofs, Solutions of x^3+y^3+z^3=nxyz, Acta Arith. LXXIII.3 (1995).
Erik Dofs and Nguyen Xuan Tho, The equation x_1/x_2 + x_2/x_3 + x_3/x_4 + x_4/x_1 = n, Int'l J. Num. Theory (2021). See also on ResearchGate.
David J. Rusin, For which values of n is a/b + b/c + c/a = n solvable? [Archive Machine link]
David J. Rusin, For which values of n is a/b + b/c + c/a = n solvable? [Cached copy of html wrapper for paper but in pdf format (so none of the links work)]
David J. Rusin, For which values of n is a/b + b/c + c/a = n solvable? [Cached copy of .txt file]
EXAMPLE
{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]}
{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]}
{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]}
41 appears in the sequence because we can write 41 = (1^3 + 2^3 + 9^3)/(1*2*9).
For n = 142, {x,y,z} = {6587432496387235561093636933115859813174, 53881756527432415186060525094013536917351, 222932371699623861287567763383948430761525}.
CROSSREFS
Cf. A085705.
Sequence in context: A140584 A085705 A187417 * A167384 A112649 A050083
KEYWORD
nonn,nice,hard
AUTHOR
Tadaaki Ohno (t-ohno(AT)hyper.ocn.ne.jp), Aug 07 2002
EXTENSIONS
More terms from David J. Rusin, Jul 26 2003
a(52)-a(58) from Jinyuan Wang, Jul 27 2021
a(59) from Jinyuan Wang, Aug 31 2023
STATUS
approved