A191763 Integers that cannot be partitioned into a sum of an odd square, an even square and a triangular number.

3, 21, 36, 78, 105, 171, 210, 351, 465, 528, 666, 903, 990, 1176, 1275, 1485, 1596, 1953, 2346, 2628, 2775, 3081, 3570, 3741, 4095, 4278, 4656, 4851, 5253, 6105, 6555, 6786, 7260, 8256, 8778, 9045, 9591, 9870

Offset

1,1

Comments

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).

Links

Table of n, a(n) for n=1..38.

Zhi-Wei Sun, Mixed sums of squares and triangular numbers, Acta Arith. 127, (2007), No. 2, pp. 103-113.

Example

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.

Mathematica

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 &]

Prog

(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

Crossrefs

Cf. A016754, A016742, A000217.

Sequence in context: A213141 A075732 A087690 * A227241 A076169 A298035

Adjacent sequences: A191760 A191761 A191762 * A191764 A191765 A191766

Keyword

nonn,easy

Author

Ant King, Jun 22 2011

Status

approved