A075768 - OEIS

%I #19 Sep 26 2013 08:40:39

%S 4,326,406,627,740,880,888,1026,1110,1284,1510,1528,2013,2072,3216,

%T 3260,3912,4866,4946,5064,5064,5829,7248,9768,10536,10686,11836,12122,

%U 13066,13398,13986,14248,14397,15000,15000,15430,15504,15544,15544,18582,18678

%N A Wallis pair (x,y) satisfies sigma(x^2) = sigma(y^2); sequence gives x'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="/A075768/b075768.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, 1]] (* _Jean-François Alcover_, Sep 26 2013 *)

%Y Cf. A075769, 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 Offset corrected by _Donovan Johnson_, Sep 18 2013