cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A072186 A Wallis pair (x,y) satisfies sigma(x^2) = sigma(y^2); sequence gives y's for Wallis pairs with x < y (ordered by values of x).

Original entry on oeis.org

5, 15, 35, 45, 55, 65, 85, 95, 105, 115, 135, 145, 155, 165, 185, 195, 205, 215, 235, 245, 255, 265, 285, 295, 305, 315, 335, 345, 355, 365, 385, 395, 405, 407, 415, 435, 445, 455, 465, 485, 495, 505, 489, 515, 535, 545, 555, 565, 585, 595, 605, 615, 635
Offset: 1

Views

Author

N. J. A. Sloane, Oct 19 2002

Keywords

Comments

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

Examples

			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), ...

References

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

Links

Crossrefs

Cf. A072182, A075768, A075769.

Programs

  • Haskell
    a072186 n = a072186_list !! (n-1)
-- a072186_list defined in A072182.  -- Reinhard Zumkeller, Sep 18 2013
  • Mathematica
    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, 2]] (* Jean-François Alcover, Oct 01 2019 *)
  • PARI
    {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][2],","))}

Extensions

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

Views

Author

N. J. A. Sloane, Oct 13 2002

Keywords

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

Crossrefs

Cf. A075769, 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, 1]] (* Jean-François Alcover, Sep 26 2013 *)

Extensions

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

Views

Author

N. J. A. Sloane, Oct 13 2002

Keywords

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

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

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

Views

Author

Klaus Brockhaus, Oct 22 2002

Keywords

Comments

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

Links

Crossrefs

Cf. A072182, A072186, A075768, A075769.

Formula

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

Extensions

Offset corrected by Donovan Johnson, Sep 18 2013
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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