A309388 Numbers y such that x*(x+1) + y*(y+1) = z*(z+1) does not have a solution in positive integers x, z with x <= y.

1, 3, 4, 7, 8, 11, 12, 15, 16, 19, 23, 28, 31, 32, 36, 40, 43, 47, 52, 59, 60, 63, 64, 67, 71, 72, 79, 83, 87, 88, 96, 100, 103, 107, 108, 112, 127, 128, 131, 136, 139, 148, 151, 156, 163, 167, 172, 176, 179, 180, 183, 187, 191, 192, 196, 199, 211, 223, 227

Offset: 1

Author

Ralf Steiner, Aug 02 2019

Keywords

• nonn

Comments

The similar sequence A027861 (complement of A012132) is related to primes.

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..10000 (n = 1..1000 from Robert Israel)

Crossrefs

Complement of A308395.

Cf. A012132, A027861.

Programs

Maple

filter:= proc(y) local S;
  S:= map(t -> subs(t, x), [isolve(x*(x+1)+y*(y+1)=z*(z+1))]);
  select(t -> t>0 and t<=y, S) = []
end proc:
select(filter, [$1..300]); # Robert Israel, Aug 06 2019

Mathematica

max = 500; lst = {}; For[x = 1, x < max, x++,
For[y = x, y < max, y++,
  For[z = y, z < max, z++,
   If[x (x + 1) + y (y + 1) == z (z + 1),
    lst = AppendTo[lst, y]]]]]; lst =
Select[Union[lst], # < max/2 &]; Complement[Range[Length[lst]], lst]

Python

from sympy import integer_nthroot
A309388_list, y, w = [], 1, 0
while len(A309388_list) < 10000:
    w += y
    z = 0
    for x in range(1,y+1):
        z += x
        if integer_nthroot(8*(w+z)+1,2)[1]:
            break
    else:
        A309388_list.append(y)
    y += 1 # Chai Wah Wu, Aug 07 2019