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

Author

César Eliud Lozada, Aug 10 2020

Keywords

• nonn
• easy

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

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

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