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A072716 Integers expressible as (x^3 + y^3 + z^3)/xyz 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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

It is easy to see that the numbers of the form m^2 + 5 are included in this sequence for n = 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 'Mma lines' for generating expressions): n = (m^2 + 33)/2: 17, 21, 29, 41, 57, 77, 101, 129, 161, 197, 237, 281, 329, 381, 437, 497, ...; n = m^4 + 8*m^2 + 4*m + 13: ..., 818, 381, 154, 53, 18, 13, 26, 69, 178, 413, 858, ...; n = 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

Table of n, a(n) for n=1..58.

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 [https://www.researchgate.net/profile/Tho-Nguyen-18/publication/350958036_The_Diophantine_equation_x_1x_2x_2x_3x_3x_4x_4x_1n/ Abstract].

David J. Rusin, For which values of n is a/b + b/c + c/a = n solvable? [Broken 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

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}.

MATHEMATICA

{n, x, y, z} = {(k^2 + 33)/2, {(k^4 + 6k^3 + 36k^2 + 98k + 147)/16, (k^4 - 6k^3 + 36k^2 - 98k + 147)/16, (k^2 + 147)/4}/GCD[(k^4 + 6k^3 + 36k^2 + 98k + 147)/16, (k^4 - 6k^3 + 36k^2 - 98k + 147)/16]}

{n, x, y, z} = {k^4 + 8k^2 + 4k + 13, {k^6 + k^5 + 10k^4 + 11k^3 + 28k^2 + 27k + 13, k^6 - 3k^5 + 12k^4 - 19k^3 + 30k^2 - 21k + 9, 2k^2 - 2k + 38}/ GCD[k^6 + k^5 + 10k^4 + 11k^3 + 28k^2 + 27k + 13, k^6 - 3k^5 + 12k^4 - 19k^3 + 30k^2 - 21k + 9]}

{n, x, y, z} = {k^4 - 2k^3 + 4k^2 + 3, {k^4 - 3k^3 + 6k^2 - 5k + 3, k^2 - k + 3, k^2 - 3k + 3}/GCD[k^2 - k + 3, k^2 - 3k + 3]}

CROSSREFS

Cf. A085705.

Sequence in context: A140584 A085705 A187417 * A167384 A112649 A050083

Adjacent sequences:  A072713 A072714 A072715 * A072717 A072718 A072719

KEYWORD

nonn,nice,more

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

STATUS

approved

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Last modified September 27 19:17 EDT 2022. Contains 357062 sequences. (Running on oeis4.)