primes. Moreover, for all sufficiently large , the interval contains many practical numbers. Let count how many practical numbers are at conjectured that is asymptotic to for some constant , a formula which resembles the prime number theorem, strengthening the earlier claim of that the practical numbers have density zero in the integers. Improving on an estimate of , found that has order of magnitude . proved Margenstern's conjecture. We have where Thus the practical numbers are about 33.6% more numerous than the prime numbers. The exact value of the constant factor is given by where is the Euler–Mascheroni constant and runs over primes. As with prime numbers in an arithmetic progression, given two natural numbers and , we have The constant factor is positive if, and only if, there is more than one practical number congruent to . If , then . For example, about 38.26% of practical numbers have a last decimal digit of 0, while the last digits of 2, 4, 6, 8 each occur with the same relative frequency of 15.43%. Notes References . . As cited by . . . As cited by . . . . . . . As cited by and . . . . . . . . . . . . . . External links Tables of practical numbers compiled by Giuseppe Melfi. Integer sequences Egyptian fractions