On generalizations of problems of Recaman and Pomerance

Answering a question of Balasubramanian, we find all primes p for which there exist p consecutive primes forming a complete residue system (mod p). On the other hand, under the prime ℓ-tuple conjecture we show that for any k≥2, there exist infinitely many sets of φ(k) consecutive primes forming reduced residue classes (mod k). The problems considered are generalizations of those of Recaman and Pomerance, respectively.