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 1-3 of 3 results.

A025052 Numbers not of form ab + bc + ca for 1<=a<=b<=c (probably the list is complete).

Original entry on oeis.org

1, 2, 4, 6, 10, 18, 22, 30, 42, 58, 70, 78, 102, 130, 190, 210, 330, 462
Offset: 1

Views

Author

Clark Kimberling

Keywords

Comments

According to Borwein and Choi, if the Generalized Riemann Hypothesis is true, then this sequence has no larger terms, otherwise there may be one term greater than 10^11. - T. D. Noe, Apr 08 2004
Note that n+1 must be prime for all n in this sequence. - T. D. Noe, Apr 28 2004
Borwein and Choi prove (Theorem 6.2) that the equation N=xy+xz+yz has an integer solution x,y,z>0 if N contains a square factor and N is not 4 or 18. In the following simple proof explicit solutions are given. Let N=mn^2, m,n integer, m>0, n>1. If n3: x=6, y=n-3, z=n^2-4n+6. If n>m+1: if n=0 (mod m+1): x=m+1, y=m(m+1), z=m(n^2/(m+1)^2-1), if n=k (mod m+1), 0

Links

Crossrefs

Subsequence of A000926 (numbers not of the form ab+ac+bc, 0A006093.
Cf. A027563, A027564, A027565, A027566, A055745, A034168, A094379, A094380, A094381.
Cf. A093669 (numbers having a unique representation as ab+ac+bc, 0A093670 (numbers having a unique representation as ab+ac+bc, 0<=a<=b<=c).

Programs

  • Mathematica
    n=500; lim=Ceiling[(n-1)/2]; lst={}; Do[m=a*b+a*c+b*c; If[m<=n, lst=Union[lst, {m}]], {a, lim}, {b, lim}, {c, lim}]; Complement[Range[n], lst]

Extensions

Corrected by R. H. Hardin

A034169 Disjoint discriminants (one form per genus) of type 2.

Original entry on oeis.org

1, 3, 5, 11, 15, 21, 29, 35, 39, 51, 65, 95, 105, 165, 231
Offset: 1

Views

Author

Jonathan Borwein (jborwein(AT)cecm.sfu.ca), choi(AT)cecm.sfu.ca (Stephen Choi)

Keywords

References

  • J. M. Borwein, D. H. Bailey and R. Girgensohn, Experimentation in Mathematics, A K Peters, Ltd., Natick, MA, 2004. x+357 pp. See p. 198.
  • J. M. Borwein and P. B. Borwein, Pi and the AGM, page 293.
  • L. E. Dickson, Introduction to the theory of numbers, Dover, NY, 1929.

Links

Crossrefs

Cf. A034168, A034170.

Formula

a(n) = A034168(n) / 2. - Sean A. Irvine, Aug 03 2020

A055745 Squarefree numbers not of form ab + bc + ca for 1 <= a <= b <= c (probably the list is complete).

Original entry on oeis.org

1, 2, 6, 10, 22, 30, 42, 58, 70, 78, 102, 130, 190, 210, 330, 462
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

References

  • Maohua Le, A note on positive integer solutions of the equation xy+yz+zx=n, Publ. Math. Debrecen 52 (1998) 159-165; Math. Rev. 98j:11016.

Links

Crossrefs

Cf. A034168, A025052, A034169.

Programs

  • Mathematica
    solQ[n_, x_] := Reduce[1 <= y <= z && n == x*y + y*z + z*x, {y, z}, Integers] =!= False; solQ[n_] := Catch[xm = Ceiling[(n-1)/2]; For[x = 1, x <= xm, x++, Which[ solQ[n, x] === True, Throw[True], x == xm, Throw[False]]]] ; solQ[1] = False; Reap[ Do[ If[ SquareFreeQ[n], If[! solQ[n] , Print[n]; Sow[n]]], {n, 1, 500}]][[2, 1]] (* Jean-François Alcover, Jun 15 2012 *)
Showing 1-3 of 3 results.

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

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