Prime power

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In mathematics, a prime power is a positive integer power of a prime number. For example: 5=5¹, 9=3² and 16=2⁴ are prime powers, while 6, 15 and 36 are not. The twenty smallest prime powers are (sequence A000961 in OEIS):

2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, ...

The prime powers are those positive integers that are divisible by just one prime number.

A property of prime powers used frequently in analytic number theory is that the set of prime powers which are not prime is a small set in the sense that the infinite sum of their reciprocals converges, although the primes are a large set.

The number of elements of a finite field is always a prime power and conversely, every prime power occurs as the number of elements in some finite field (which is unique up to isomorphism).

The divisor function calculated on a prime power can be calculated with the formula

$\sigma(p^n)\;=\;\frac{1 - p^{n+1}}{1 - p}$

verifiable by using the formula for the sum of a geometric series. All prime powers are deficient numbers. A prime power pⁿ is an n-almost prime.

It is not known whether a prime power pⁿ can be an amicable number. If there is such a number, then pⁿ must be greater than 10¹⁵⁰⁰ and n must be greater than 1400.