%N Numbers which are not the sum of a triangular number and zero or a prime = 3 (mod 4).
%C Zhi-Wei SUN conjectured that n=216 is the only number not of the form n = p + k(k+1)/2 with p a prime or zero and more precisely that the numbers given in this sequence A137997(1..112) = 2,...,90441 are the only numbers which cannot be written in this form with p=0 or p=3 (mod 4).
%H Zhi-Wei SUN, <a href="https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;9ebd2e77.0803">A new conjecture: n=p+x(x+1)/2</a>, Mar 23, 2008.
%o (PARI) for( n=1,10^8, t=sqrtint( 2*n ); if( 0>= p = n-t*(t+1)/2, !p && next, p -= t++ ); until( !t--, isprime( p+=t ) || next; p%4==3 && next(2)); print1( n"," ))
%Y Cf. A137996, A132399, A000040, A000217.
%A _M. F. Hasler_, Mar 24 2008