  We study whether sufficiently large integers can be written in the form
cp+T_x, where p is either zero or a prime congruent to r mod d, and
T_x=x(x+1)/2 is a triangular number. We also investigate whether there are
infinitely many positive integers not of the form (2^ap-r)/m+T_x with p a prime
and x an integer. Besides two theorems, the paper also contains several
conjectures together with related analysis and numerical data. One of our
conjectures states that each natural number not equal to 216 can be written in
the form p+T_x with x an integer and p a prime or zero; another conjecture
asserts that any odd integer n>3 can be written in the form p+x(x+1) with p a
prime and x a positive integer.
