structural:summarization, linguistic:paraphrasing
Prime factors

For large values of n, the n -th telephone number is divisible by a large power of two, 2. More precisely, the 2-adic order (the number of factors of two in the prime factorization) of T (4 k) and of T (4 k + 1) is k; for T (4 k + 2) it is k + 1, and for T (4 k + 3) it is k + 2.

For any prime number p, one can test whether there exists a telephone number divisible by p by computing the recurrence for the sequence of telephone numbers, modulo p, until either reaching zero or detecting a cycle. The primes that divide at least one telephone number are

The odd primes in this sequence have been called inefficient. Each of them divides infinitely many telephone numbers.