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

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

1,1

Comments

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

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Example

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.

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<n and k>=2 is 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 *)

Prog

(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

Crossrefs

Subsequence of A336607.

Sequence in context: A062486 A193226 A291555 * A053698 A162867 A336971

Adjacent sequences: A336992 A336993 A336994 * A336996 A336997 A336998

Keyword

nonn,easy

Author

César Eliud Lozada, Aug 10 2020

Status

approved