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A191764
Integers that do not have a partition into a sum of an odd square and two (not necessarily distinct) triangular numbers.
1
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
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
KEYWORD
nonn,easy
AUTHOR
Ant King, Jun 22 2011
STATUS
approved