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A075768 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). 5
4, 326, 406, 627, 740, 880, 888, 1026, 1110, 1284, 1510, 1528, 2013, 2072, 3216, 3260, 3912, 4866, 4946, 5064, 5064, 5829, 7248, 9768, 10536, 10686, 11836, 12122, 13066, 13398, 13986, 14248, 14397, 15000, 15000, 15430, 15504, 15544, 15544, 18582, 18678 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois Alcover, Sep 26 2013 *)

CROSSREFS

Cf. A075769, A072182, A072186, A077053.

Sequence in context: A053917 A005832 A195501 * A135442 A086895 A293241

Adjacent sequences:  A075765 A075766 A075767 * A075769 A075770 A075771

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

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Last modified July 22 06:33 EDT 2019. Contains 325216 sequences. (Running on oeis4.)