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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.

%I A309388 #39 Aug 08 2019 02:59:35
%S A309388 1,3,4,7,8,11,12,15,16,19,23,28,31,32,36,40,43,47,52,59,60,63,64,67,
%T A309388 71,72,79,83,87,88,96,100,103,107,108,112,127,128,131,136,139,148,151,
%U A309388 156,163,167,172,176,179,180,183,187,191,192,196,199,211,223,227
%N 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.
%C A309388 The similar sequence A027861 (complement of A012132) is related to primes.
%H A309388 Chai Wah Wu, <a href="/A309388/b309388.txt">Table of n, a(n) for n = 1..10000</a> (n = 1..1000 from Robert Israel)
%p A309388 filter:= proc(y) local S;
%p A309388   S:= map(t -> subs(t, x), [isolve(x*(x+1)+y*(y+1)=z*(z+1))]);
%p A309388   select(t -> t>0 and t<=y, S) = []
%p A309388 end proc:
%p A309388 select(filter, [$1..300]); # _Robert Israel_, Aug 06 2019
%t A309388 max = 500; lst = {}; For[x = 1, x < max, x++,
%t A309388 For[y = x, y < max, y++,
%t A309388   For[z = y, z < max, z++,
%t A309388    If[x (x + 1) + y (y + 1) == z (z + 1),
%t A309388     lst = AppendTo[lst, y]]]]]; lst =
%t A309388 Select[Union[lst], # < max/2 &]; Complement[Range[Length[lst]], lst]
%o A309388 (Python)
%o A309388 from sympy import integer_nthroot
%o A309388 A309388_list, y, w = [], 1, 0
%o A309388 while len(A309388_list) < 10000:
%o A309388     w += y
%o A309388     z = 0
%o A309388     for x in range(1,y+1):
%o A309388         z += x
%o A309388         if integer_nthroot(8*(w+z)+1,2)[1]:
%o A309388             break
%o A309388     else:
%o A309388         A309388_list.append(y)
%o A309388     y += 1 # _Chai Wah Wu_, Aug 07 2019
%Y A309388 Complement of A308395.
%Y A309388 Cf. A012132, A027861.
%K A309388 nonn
%O A309388 1,2
%A A309388 _Ralf Steiner_, Aug 02 2019