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%I #16 Sep 26 2013 08:40:49
%S 5,407,489,749,878,1451,1102,1208,1943,1528,1809,1605,2557,3097,3730,
%T 4829,6061,4880,6341,6172,7715,7067,10071,17441,11020,17531,14397,
%U 17441,14001,24161,24613,14288,14795,20396,25495,22577,19784,15836,19795,27713,30959
%N A Wallis pair (x,y) satisfies sigma(x^2) = sigma(y^2); sequence gives y's for indecomposable Wallis pairs with x < y (ordered by values of x).
%C 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.
%D I. Kaplansky, The challenges of Fermat, Wallis and Ozanam (and several related challenges): II. Fermat's second challenge, Preprint, 2002.
%H Donovan Johnson, <a href="/A075769/b075769.txt">Table of n, a(n) for n = 1..1000</a>
%e (4,5) is a Wallis pair since sigma(16) = sigma(25) = 31.
%t 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, 2]] (* _Jean-François Alcover_, Sep 26 2013 *)
%Y Cf. A075768, A072182, A072186, A077053.
%K nonn,nice
%O 1,1
%A _N. J. A. Sloane_, Oct 13 2002
%E Corrected and extended by _Klaus Brockhaus_, Oct 22 2002
%E 19795 from _Jean-François Alcover_, Dec 28 2012
%E Offset corrected by _Donovan Johnson_, Sep 18 2013
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