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 1-2 of 2 results.

A336995 Numbers of the form x^3 + x^2*y + x*y^2 + y^3, where x and y are coprime positive integers.

Original entry on oeis.org

4, 15, 40, 65, 85, 156, 175, 203, 259, 272, 369, 400, 477, 580, 585, 671, 715, 803, 820, 888, 935, 1105, 1111, 1157, 1261, 1417, 1464, 1484, 1625, 1695, 1820, 1885, 2055, 2080, 2336, 2380, 2465, 2533, 2595, 2669, 2848, 2873, 2955, 3060, 3145, 3439, 3485, 3492
Offset: 1

Views

Author

César Eliud Lozada, Aug 10 2020

Keywords

Comments

Equivalently, numbers of the form (x+y)*(x^2 + y^2) where x and y are coprime positive integers.

Examples

			For x=1, y=1, x^3+x^2*y+x*y^2+y^3 = 4, so 4 is in the sequence.
For x=1, y=2, x^3+x^2*y+x*y^2+y^3 = 15, so 15 is in the sequence.
For x=2, y=2, x^3+x^2*y+x*y^2+y^3 = 32, but GCD(x,y)<>1, so 32 is not in the sequence.

Links

Crossrefs

Subsequence of A336607.

Programs

  • Maple
    N:= 10000: # for terms <= N
S:= {}:
for x from 1 while (x+1)*(x^2+1) < N do
   V:= select(`<=`,map(y -> (x+y)*(x^2+y^2), select(y -> igcd(x,y)=1, {seq(i,i=1..min(x,(N-x^3)/x^2))})),N);
   S:= S union V;
od:
sort(convert(S,list)); # Robert Israel, Sep 21 2020
  • Mathematica
    max = 5000; T0 = {}; xm = Ceiling[Sqrt[max]];
While[T = T0;
T0 = Table[x^3 + x^2 y + x y^2 + y^3, {x, 1, xm}, {y, x, xm}] //
     Flatten // Union // Select[#, # <= max &] &; T != T0, xm = 2 xm];
T (* T=A336607 *)
(* Now, exclude a(n) such that a(n)=k^3*a(m) for m=2 is an integer *)
T2 = T; n = 1;
While[n <= Length[T2],
  t1 = T2[[n]]; t2 = Last[T2]; max2 = 1 + (t2/t1)^(1/3);
  T2 = Complement[T2, Table[t1*k^3, {k, 2, max2}]];
  n++;
  ];
T2 (* T2=A336995 *)
  • PARI
    upto(limit)={my(L=List(), b=sqrtnint(limit,3)); for(x=1, b, for(y=1, b, my(t=(x+y)*(x^2+y^2)); if(t<=limit && gcd(x,y)==1, listput(L,t)) )); Set(L)}
upto(4000) \\ Andrew Howroyd, Aug 10 2020

A337786 Numbers of the form x^3 + x^2*y + x*y^2 + y^3 where x and y are positive integers, but having no such representation where x and y are coprime.

Original entry on oeis.org

32, 108, 120, 256, 320, 405, 500, 520, 680, 864, 960, 1080, 1248, 1372, 1400, 1624, 1755, 1875, 2048, 2072, 2176, 2295, 2560, 2916, 2952, 3200, 3240, 3816, 4000, 4160, 4212, 4640, 4680, 4725, 5000, 5145, 5324, 5368, 5440, 5481, 5720, 6424, 6560, 6912, 6993, 7104, 7344, 7480, 7680, 8125, 8640
Offset: 1

Views

Author

Robert Israel, Sep 21 2020

Keywords

Comments

Complement of A336995 in A336607.

Examples

			a(3)=120 is a member because 120 = x^3 + x^2*y + x*y^2 + y^3 where x=2 and y=4, but has no such representation where x and y are coprime positive integers.
206312 is not a member because although 206312 = x^3 + x^2*y + x*y^2 + y^3 where x=32 and y=42 are not coprime, it also has such a representation where x=15 and y=53 are coprime.

Links

Crossrefs

Cf. A336607, A336995. Contained in A046099.

Programs

  • Maple
    N:= 10000: # for terms <= N
S1:= {}: S2:= {}:
for x from 1 while (x+1)*(x^2+1) < N do
   C:= {seq(i,i=1..min(x,(N-x^3)/x^2))}:
   C1,C2:= selectremove(y -> igcd(x,y)=1, C);
   V1:= select(`<=`,map(y -> (x+y)*(x^2+y^2), C1),N);
   V2:=  select(`<=`,map(y -> (x+y)*(x^2+y^2), C2),N);
   S1:= S1 union V1;
   S2:= S2 union V2;
od:
sort(convert(S2 minus S1,list));
Showing 1-2 of 2 results.

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