On Graham partitions twisted by the Legendre symbol

We investigate when there is a partition of a positive integer n n, n = f (λ 1) + f (λ 2) + + f (λ ), n=f\left({\lambda }_{1})+f\left({\lambda }_{2})+\cdots +f\left({\lambda }_{\ell }), satisfying that 1 = χ p (λ 1) λ 1 + χ p (λ 2) λ 2 + + χ p (λ ) λ , 1=\frac{{\chi }_{p}\left({\lambda }_{1})}{{\lambda }_{1}}+\frac{{\chi }_{p}\left({\lambda }_{2})}{{\lambda }_{2}}+\cdots +\frac{{\chi }_{p}\left({\lambda }_{\ell })}{{\lambda }_{\ell }}, where χ p {\chi }_{p} is the Legendre symbol modulo prime p p and f (k) = k f\left(k)=k or the k k th m m -gonal number with m = 3 m=3, 4, or 5.