cp's OEIS Frontend

Let H(n,b) be the Hamming graph whose vertices are the sequences of length n over the alphabet {0,1,...,b-1} with adjacency being defined by having Hamming distance 1. Let P(n,b) be the subgraph of H(n,b) induced by the set of vertices which are base b representations of primes with n digits (not allowing leading 0 digits).

For numerator of polynomial of genus 1 and level n for m = 1 see A001008.

Definition: The polynomial A[1,2n+1](m) = A[genus 1,level n] is here defined as

Sum_{d,1,n-1} m^(n-d)/d.

Few first A[1,n](m):

n=1: A[1,1](m)= 0;

n=2: A[1,2](m)= m;

n=3: A[1,3](m)= m/2 + m^2;

n=4: A[1,4](m)= m/4 + m^2/3 + m^3/2 + m^4.

General formula which uses these polynomials is following:

(1/(n+1))Hypergeometric2F1[1,n,n+1,1/m] =

Sum_{x>=0} m^(-x)/(x+n) =

m^n*arctanh((2m-1)/(2m^2-2m+1)) - A[1,n](m) =

m^n*log(m/(m-1)) - A[1,n](m).