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

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