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A075768
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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).
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5
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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
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OFFSET
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1,1
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COMMENTS
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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.
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REFERENCES
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I. Kaplansky, The challenges of Fermat, Wallis and Ozanam (and several related challenges): II. Fermat's second challenge, Preprint, 2002.
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LINKS
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EXAMPLE
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(4,5) is a Wallis pair since sigma(16) = sigma(25) = 31.
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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