A191764 Integers that do not have a partition into a sum of an odd square and two (not necessarily distinct) triangular numbers.

6, 42, 72, 156, 210, 342, 420, 702, 930, 1056, 1332, 1806, 1980, 2352, 2550, 2970, 3192, 3906, 4692, 5256, 5550, 6162, 7140, 7482, 8190, 8556, 9312, 9702, 10506, 12210, 13110, 13572, 14520, 16512, 17556, 18090, 19182, 19740, 20880, 21462, 23256, 24492, 25122, 26406, 28392, 30450, 31152, 33306, 34782, 35532, 37830

Offset

1,1

Comments

Oh & Sun have proved that a natural number cannot be partitioned into a sum of an odd square and two triangular numbers if and only if it is a pronic number A002378 (m) such that 2m+1 does not have any prime divisors that are congruent to 3 (mod 4).

Links

Donovan Johnson, Table of n, a(n) for n = 1..1000

Byeong-Kweon Oh and Zhi-Wei Sun, Mixed sums of square and triangular numbers (III), Journal of Number Theory 129:4, (2009), pp. 964-969.

Example

The fifth integer that does not have a partition into a sum of an odd square and two triangular numbers is 210. Hence a(5)=210. Similarly, 21 is not in the sequence as it has a unique representation as A000290(3)+A000217(3)+A000217(3)

Mathematica

data=Length[FindInstance[(2x+1)^2+1/2 y (y+1)+1/2 z (z+1)==# && 0<=x<=# && 0<=y<=# && 0<=z<=#, {x, y, z}, Integers]]&/@Range[10000]; Position[data, 0]//Flatten

Prog

(PARI)
N=10^5; /* upper bound */
x='x+O('x^N);
S=2*ceil(sqrt(N));
tr=sum(n=0, S, x^(n*(n+1)/2)); /* triangular incl. zero */
sq=sum(n=1, S, x^((2*n-1)^2)); /* odd squares */
f=tr^2*sq + 't /* symbol t to have vector aligned */
v=Vec(f);
for(n=1, #v, if(v[n]==0, print1(n-1, ", ")));
/* Joerg Arndt, Jul 06 2011 */

Crossrefs

Cf. A002378, A016754, A000217.

Sequence in context: A329339 A176308 A103763 * A043896 A044489 A211616

Adjacent sequences: A191761 A191762 A191763 * A191765 A191766 A191767

Keyword

nonn,easy

Author

Ant King, Jun 22 2011

Status

approved