A075769 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).

5, 407, 489, 749, 878, 1451, 1102, 1208, 1943, 1528, 1809, 1605, 2557, 3097, 3730, 4829, 6061, 4880, 6341, 6172, 7715, 7067, 10071, 17441, 11020, 17531, 14397, 17441, 14001, 24161, 24613, 14288, 14795, 20396, 25495, 22577, 19784, 15836, 19795, 27713, 30959

Offset: 1

Author

N. J. A. Sloane, Oct 13 2002

Keywords

• nonn
• nice

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.

Examples

			(4,5) is a Wallis pair since sigma(16) = sigma(25) = 31.
		

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

Crossrefs

Cf. A075768, A072182, A072186, A077053.

Programs

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, 2]] (* Jean-François Alcover, Sep 26 2013 *)

Extensions

Corrected and extended by Klaus Brockhaus, Oct 22 2002

19795 from Jean-François Alcover, Dec 28 2012

Offset corrected by Donovan Johnson, Sep 18 2013