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.)

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

Jinyuan Wang, Table of n, a(n) for n = 1..250

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

Adjacent sequences: A072713 A072714 A072715 * A072717 A072718 A072719

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