A072186 -id:A072186 - cp's OEIS frontend

A072182 A Wallis pair (x,y) satisfies sigma(x^2) = sigma(y^2); sequence gives x's for Wallis pairs with x < y (ordered by values of x, then y).

4, 12, 28, 36, 44, 52, 68, 76, 84, 92, 108, 116, 124, 132, 148, 156, 164, 172, 188, 196, 204, 212, 228, 236, 244, 252, 268, 276, 284, 292, 308, 316, 324, 326, 332, 348, 356, 364, 372, 388, 396, 404, 406, 412, 428, 436, 444, 452, 468, 476, 484, 492, 508, 516

Offset: 1

N. J. A. Sloane, Oct 19 2002

4*A045572 is included in this sequence. - Benoit Cloitre, Oct 22 2002

D. Johnson remarks that some terms are repeated, e.g., a(139)=a(140)=1284 forms a Wallis pair with A072186(139)=1528 and also with A072186(140)=1605. - M. F. Hasler, Sep 15 2013

			The first few pairs are all multiples of the first pair (4,5): (4, 5), (12, 15), (28, 35), (36, 45), (44, 55), (52, 65), ...

I. Kaplansky, The challenges of Fermat, Wallis and Ozanam (and several related challenges): II. Fermat's second challenge, Preprint, 2002.

Donovan Johnson, Table of n, a(n) for n = 1..10000

Cf. A072186, A075768, A075769.

Cf. A000203, A000290.

a072182 n = a072182_list !! (n-1)
(a072182_list, a072186_list) = unzip wallisPairs
  wallisPairs = [(x, y) | (y, sy) <- tail ws,
                          (x, sx) <- takeWhile ((< y) . fst) ws, sx == sy]
                where ws = zip [1..] $ map a000203 $ tail a000290_list
-- Reinhard Zumkeller, Sep 17 2013

w = {}; m = 550;
Do[q = DivisorSigma[1, x^2]; sq = Sqrt[q] // Floor; Do[If[q == DivisorSigma[1, y^2], AppendTo[w, {x, y}]], {y, x+1, sq}], {x, 1, m}];
w[[All, 1]] (* Jean-François Alcover, Oct 01 2019 *)

{w=[]; m=550; for(x=1,m,q=sigma(x^2); sq=sqrtint(q); for(y=x+1,sq,if(q==sigma(y^2), w=concat(w,[[x,y]])))); for(j=1,matsize(w)[2],print1(w[j][1],","))}

Extended by Klaus Brockhaus and Benoit Cloitre, Oct 22 2002

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).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Oct 13 2002

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.

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

I. Kaplansky, The challenges of Fermat, Wallis and Ozanam (and several related challenges): II. Fermat's second challenge, Preprint, 2002.

Donovan Johnson, Table of n, a(n) for n = 1..1000

Cf. A075769, A072182, A072186, A077053.

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 *)

Corrected and extended by Klaus Brockhaus, Oct 22 2002

Offset corrected by Donovan Johnson, Sep 18 2013

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).

Original entry on oeis.org

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

N. J. A. Sloane, Oct 13 2002

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.

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

I. Kaplansky, The challenges of Fermat, Wallis and Ozanam (and several related challenges): II. Fermat's second challenge, Preprint, 2002.

Donovan Johnson, Table of n, a(n) for n = 1..1000

Cf. A075768, A072182, A072186, A077053.

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 *)

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

A077053 Greatest common divisor of indecomposable Wallis pairs.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 2, 1, 4, 1, 1, 1, 1, 2, 1, 1, 2, 1, 4, 1, 1, 3, 1, 4, 1, 1, 1, 1, 1, 1, 8, 1, 4, 5, 1, 8, 4, 1, 1, 1, 1, 1, 8, 2, 1, 1, 1, 6, 1, 8, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 1, 2, 1, 4, 2, 2, 1, 1, 4, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 6, 4

Offset: 1

Klaus Brockhaus, Oct 22 2002

nonn

Terms > 1 show that (x,y) need not be a Wallis pair if (cx,cy) is a Wallis pair.

Donovan Johnson, Table of n, a(n) for n = 1..1000

Cf. A072182, A072186, A075768, A075769.

a(n) = gcd(A075768(n), A075769(n)).

Offset corrected by Donovan Johnson, Sep 18 2013

cp's OEIS Frontend

A072182 A Wallis pair (x,y) satisfies sigma(x^2) = sigma(y^2); sequence gives x's for Wallis pairs with x < y (ordered by values of x, then y).

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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).

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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).

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A077053 Greatest common divisor of indecomposable Wallis pairs.

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