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.

A265731 Powers C^z = A^x + B^y with all positive integers and x,y,z > 1, without multiplicity.

Original entry on oeis.org

8, 9, 16, 25, 32, 36, 64, 81, 100, 125, 128, 144, 169, 196, 225, 243, 256, 289, 324, 343, 400, 441, 512, 576, 625, 676, 784, 841, 900, 1000, 1024, 1089, 1156, 1225, 1296, 1369, 1521, 1600, 1681, 1728, 1764, 1849, 2025, 2048, 2197, 2304, 2500, 2601, 2704, 2744, 2809, 2916, 3025, 3125
Offset: 1

Views

Author

Anatoly E. Voevudko, Dec 14 2015

Keywords

Comments

This type of equation is used in the Fermat-Catalan conjecture, the ABC conjecture, etc., of course, with additional restrictions and conditions.

Examples

			8 = 2^3 = 2^2 + 2^2; 9 = 3^2 = 1^3 + 2^3; 16 = 4^2 = 2^3 + 2^3; etc.

Links

Crossrefs

Cf. A000290, A245713, A261782, A264901, A265732.

Programs

  • PARI
    A265731(lim,bflag=0)={my(Lcz=List(1),Lb=List(),czn,lczn,lbn,lim2=logint(lim, 2),lim3);
for(z=2, lim2, lim3=sqrtnint(lim, z); for(C=2, lim3, listput(Lcz, C^z)));
Lcz=Set(Lcz); lczn = #Lcz; if(lczn==0,return(-1));
for(i=1, lczn, for(j=i, lczn, czn=Lcz[i]+Lcz[j]; if(czn>lim, break);
if(setsearch(Lcz, czn), listput(Lb, czn)))); listsort(Lb,1);  lbn=#Lb;
if(bflag, for(i=1,lbn,print(i , " ", Lb[i]))); if(!bflag,return(Vec(Lb))); }
\\ Anatoly E. Voevudko, Nov 23 2015

A265732 Powers C^z = A^x + B^y with all positive integers and x,y,z > 1, with multiplicity.

Original entry on oeis.org

8, 9, 16, 16, 25, 25, 32, 32, 32, 36, 36, 64, 64, 64, 81, 81, 100, 100, 100, 125, 125, 128, 128, 128, 128, 128, 128, 144, 144, 169, 196, 225, 225, 225, 225, 243, 256, 256, 256, 289, 289, 289, 324, 324, 324, 343, 400, 400, 400, 441, 512, 512, 512, 512, 512, 512, 512, 512, 512, 512, 512, 512, 512, 512, 576
Offset: 1

Views

Author

Anatoly E. Voevudko, Dec 14 2015

Keywords

Comments

We do not distinguish between the equations C^z = A^x + B^y and C^z = B^y + A^x.
This type of equation is used in the Fermat-Catalan conjecture, the ABC conjecture, etc., of course with additional restrictions and conditions.

Examples

			128 = 64 + 64 ==> 2^7 = 8^2 + 8^2 = 8^2 + 4^3 = 8^2 + 2^6 = 4^3 + 4^3 = 4^3 + 2^6 = 2^6 + 2^6 (but not 4^3 + 8^2, 2^6 + 8^2, 2^6 + 4^3).

Links

Crossrefs

Cf. A000290, A245713, A261782, A264901, A265731.

Programs

  • PARI
    A265732(lim,bflag=0)=
{my(Lc=List(1),Lb=List(),La=Lb,czn,lcn,lan,lim2=logint(lim,2),lim3,k);
for(z=2,lim2,lim3=sqrtnint(lim,z); for(C=2,lim3,listput(Lc,C^z)) );
lcn = #Lc; if(lcn==0,return(-1));
for(i=1,lcn, for(j=i,lcn, czn=Lc[i]+Lc[j]; if(czn>lim, next);
La=findinlista(Lc, czn); lan=#La; if(!lan, next);
for(k=1,lan, listput(Lb, czn)))); lcn=#Lb; listsort(Lb,0);
if(bflag,for(i=1,lcn,print(i ," ",Lb[i]))); if(!bflag,return(Vec(Lb)));
}
findinlista(list, item, sind=1)={my(ln=#list, Li=List());
if(ln==0||sind<1||sind>ln, return(Li));
for(i=sind, ln, if(list[i]==item, listput(Li,i))); return(Li);
} \\ Anatoly E. Voevudko, Nov 23 2015

A264801 Number of essentially different seating arrangements for 2n couples around a circular table with 4n seats such that no spouses are neighbors, the neighbors of each person have opposite gender and no person's neighbors belong to the same couple.

Original entry on oeis.org

0, 6, 2400, 6375600, 45927907200, 713518388352000, 21216194909362252800, 1105729617210350356224000, 94398452626533646953922560000, 12514511465855205467497303154688000, 2467490887755897725667792936979169280000, 698323914872709997998407130752506728284160000
Offset: 1

Views

Author

Hugo Pfoertner, Nov 25 2015

Keywords

Comments

This might be called the "maximum diversity" menage problem. Arrangements that differ only by rotation or reflection are excluded by the following conditions: Seat number 1 is assigned to person A. Seat number 2 can only be taken by a person of the same gender as A. The second condition forces an mmffmmff... pattern.

Examples

			a(1)=0 because with 2 couples it is impossible to satisfy all three conditions.
a(2)=6 because only the following arrangements are possible with 4 couples: ABdaCDbc, ABcaDCbd, ACdaBDcb, ACbaDBcd, ADcaBCdb, ADbaCBdc. This corresponds to the (2*2-1)! possibilities for persons B, C and D to choose a seat. After the positions of A, B, C and D are fixed, only A000183(2*2)=1 possibility remains to arrange their spouses a, b, c  and d.

Crossrefs

Cf. A000183, A007060, A094047, A114939, A258338.

Programs

  • PARI
    a000183(N)={my(a0=[0,0,0,1,2,20],a=vector(N),
f(x)=fibonacci(x-1)+fibonacci(x+1)+2;);
if(N<7,a=a0[1..N],for(k=1,6,a[k]=a0[k]);
for(n=7,N,a[n] = (-1)^n*(4*n+f(n)) +
 (n/(n-1))*((n+1)*a[n-1] + 2*(-1)^n*f(n-1))
  - ((2*n)/(n-2))*((n-3)*a[n-2] + (-1)^n*f(n-2))
  + (n/(n-3))*((n-5)*a[n-3] + 2*(-1)^(n-1)*f(n-3))
  + (n/(n-4))*(a[n-4] + (-1)^(n-1)*f(n-4))));a};
a264901(limit)={my(a183=a000183(2*limit)); for(n=1,limit,print1((2*n-1)!*a183[2*n],", "))};
a264901(12) \\ Hugo Pfoertner, Sep 05 2020

Formula

a(n) = (2*n-1)! * A000183(2*n).
Showing 1-3 of 3 results.

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