On the number of numerical semigroups \(\langle a,b\rangle\) of prime power genus. (Q372348)

Let \(a\) and \(b\) be positive integers. Then the set \(S=\langle a,b\rangle=\{xa+yb\mid x,y\in\mathbb N\}\) (with \(\mathbb N\) the set of nonnegative integers) is a submonoid of \((\mathbb N,+)\) with finite complement in \(\mathbb N\), that is, a numerical semigroup. The elements in the set \(\mathbb N\setminus S\) are known as gaps of \(S\), and its cardinality is the genus of \(S\), \(g(S)\). The aim of this paper is to give formulas for \[ n(p^k,2)=\{(a,b)\in\mathbb N^2\mid\gcd(a,b)=1,\;g(\langle a,b\rangle)=p^k\} \] with \(p\) a prime positive integer. The authors first prove that \[ n(p^k,2)=\#\{0\leq i\leq k\mid\gcd(p^i+1,2p^{k-i}+1)=1\}. \] For this reason they provide alternative expressions for \(\gcd(p^\alpha+1,2p^\beta+1)\). With this it is shown that \(n(p^k,2)\) only depends of the class of \(p\) modulo \(M(k)\), which is the product of the primes dividing \(\prod_{i=1}^k(2^{i/\gcd(i,k)}-(-1)^{k/\gcd(i,k)})\). Finally with the help of the functions \(X_{a,q}\) the authors provide formulas for \(n(p^k,2)\) for \(k\in\{9,10\}\) and revisit the cases \(k\leq 8\). For \(q\) a prime integer \(X_{a,q}\) is just the characteristic function of \(\mathbb Z\setminus(a+q\mathbb Z)\). For \(q=q_1^{e_1}\cdots q_t^{e_t}\) with \(e_i\) positive integers and \(q_i\) positive prime integers, \(i\in\{1,\ldots,k\}\), \(X_{a,q}\) is defined as \(\prod_{i=1}^kX_{a,q_i}\) (so it only takes into account the square free part of \(q\), as \(M(k)\) does). An interesting open problem is formulated in the last section of this manuscript.