OFFSET
1,1
COMMENTS
If (x,y) and (u,v) are Wallis pairs, a is from (x,y) and c is from (u,v) and gcd(a,c)=1, b is from (x,y) and d is from(u,v) and gcd(b,d)=1, then (ac,bd) is also a Wallis pair. Such pairs are called decomposable. If (x,y) and (cx,cy) are Wallis pairs then (cx,cy) is also called decomposable.
REFERENCES
I. Kaplansky, The challenges of Fermat, Wallis and Ozanam (and several related challenges): II. Fermat's second challenge, Preprint, 2002.
LINKS
Donovan Johnson, Table of n, a(n) for n = 1..1000
EXAMPLE
(4,5) is a Wallis pair since sigma(16) = sigma(25) = 31.
MATHEMATICA
xmax = 20000; sigma[n_] := sigma[n] = DivisorSigma[1, n]; WallisQ[{x_, y_}] := sigma[x^2] == sigma[y^2]; pairs = Reap[Do[Do[ If[WallisQ[{x, y}] && ! (GCD[x, y] != 1 && WallisQ[{x, y}/GCD[x, y]]), Print[{x, y}, " is a Wallis pair to be tested for indecomposability"]; Sow[{x, y}]], {y, x + 1, 2.2*x}], {x, 1, xmax}]][[2, 1]]; indecomposableQ[{x0_, y0_}] := (pf = pairs // Flatten; sx = Intersection[Most@Divisors[x0], pf]; sy = Intersection[Most@Divisors[y0], pf]; xy = Outer[List, sx, sy] // Flatten[#, 1] &; sel = Select[xy, WallisQ[#] && WallisQ[{x0, y0}/#] &]; sel == {}); Select[pairs, indecomposableQ][[All, 1]] (* Jean-François Alcover, Sep 26 2013 *)
CROSSREFS
KEYWORD
nonn,nice
AUTHOR
N. J. A. Sloane, Oct 13 2002
EXTENSIONS
Corrected and extended by Klaus Brockhaus, Oct 22 2002
Offset corrected by Donovan Johnson, Sep 18 2013
STATUS
approved