An integer
is -balanced
for
a prime if, among all nonzero binomial coefficients for , ..., (mod ), there are equal numbers of quadratic residues and nonresidues
(mod ).
Let
be the set of integers , , that are -balanced. Among all the primes , only those with , 3, and 11 have .
The following table gives the -balanced integers for small primes (OEIS A093755).