cp's OEIS Frontend

These are the repunit primes in base -100. It is unusual to represent numbers in a negative base, but it follows the same formulation as any base: numbers are represented as a sum of powers in that base, i.e., a0*1 + a1*b^1 + a2*b^2 + a3*b^3 ... Since the base is negative, the terms will be alternating positive/negative. For repunits the coefficients are all ones so the sum reduces to 1 + b + b^2 + b^3 + ... + b^(k-1) = (b^k-1)/(b-1). Since b is negative and k is an odd prime, the sum equals (|b|^k+1)/(|b|+1). For k=3, the sum is 9901, which is prime. As with all repunits, we only need to PRP test the prime exponents. The factors of repunits base -100 will be of the form p=2*k*m+1 where m must be even, which is common for (negative) bases that are squares.

These are the repunit primes in base 35.

These are the Repunits in base -81. Since 81=3^4, factors will be of the form p=8nk+1. (Negative) bases that are powers of small numbers appear to have a higher frequency of primes than Repunits in other bases. The best linear fit for this base is currently 0.29918 which is much lower (better) than the conjectured 0.56145948 (see link to conjecture).