The p-Numerical Semigroup of the Triple of Arithmetic Progressions

For given positive integers a1,a2,⋯,ak with gcd(a1,a2,⋯,ak)=1, the denumerant d(n)=d(n;a1,a2,⋯,ak) is number of nonnegative solutions (x1,x2,⋯,xk) linear equation a1x1+a2x2+⋯+akxk=n for a integer n. p, let Sp=Sp(a1,a2,⋯,ak) be set all n’s such that d(n)>p. In this paper, by introducing p-numerical semigroup, where N0\Sp finite, we give explicit formulas p-Frobenius number, which maximum N0\Sp, and related values triple arithmetic progressions. The main aim to determine elements p-Apéry set.