A191763 - OEIS

%I #13 Sep 05 2018 02:28:36

%S 3,21,36,78,105,171,210,351,465,528,666,903,990,1176,1275,1485,1596,

%T 1953,2346,2628,2775,3081,3570,3741,4095,4278,4656,4851,5253,6105,

%U 6555,6786,7260,8256,8778,9045,9591,9870

%N Integers that cannot be partitioned into a sum of an odd square, an even square and a triangular number.

%C Sun has proved that the a(n) are those positive triangular numbers A000217(m) for which all the prime divisors of 2m+1 are congruent to 1 (mod 4).

%H Zhi-Wei Sun, <a href="http://dx.doi.org/10.4064/aa127-2-1">Mixed sums of squares and triangular numbers</a>, Acta Arith. 127, (2007), No. 2, pp. 103-113.

%e The fifth integer that cannot be partitioned into a sum of an odd square, an even square and a triangular number is 105. Hence a(5)=105.

%t Test[n_] := Module[{x, y, z}, FindInstance[(2x+1)^2 + (2 y)^2 + z (z+1)/2 == n && 0 <= x <= n && 0 <= y <= n && 0 <= z <= n, {x, y, z}, Integers]]; Select[Range[1000], Length[Test[#]] == 0 &]

%o (PARI) is_A191763(N)=issquare(N*8+1,&N)&N%4==1&vecsort(factor(N)[,1]~%4,,8)==[1]  \\ _M. F. Hasler_, Jun 22 2011

%Y Cf. A016754, A016742, A000217.

%K nonn,easy

%O 1,1

%A _Ant King_, Jun 22 2011