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All integers are either even or odd. A square has even multiplicity for all prime factors (it is of the form a 2 for some a). The first: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144 (sequence A000290 in the OEIS). A cube has all multiplicities divisible by 3 (it is of the form a 3 for some a).
A Pillai prime, [9] because 23! + 1 is divisible by 79, but 79 is not one more than a multiple of 23. A regular prime. [10] A right-truncatable prime, because when the last digit (9) is removed, the remaining number (7) is still prime. A sexy prime (with 73). The n value of the Wagstaff prime 201487636602438195784363.
The Goldbach conjecture verification project reports that it has computed all primes smaller than 4×10 18. [2] That means 95,676,260,903,887,607 primes [3] (nearly 10 17), but they were not stored. There are known formulae to evaluate the prime-counting function (the number of primes smaller than a given value) faster than computing the primes.
The same prime factor may occur more than once; this example has two copies of the prime factor When a prime occurs multiple times, exponentiation can be used to group together multiple copies of the same prime number: for example, in the second way of writing the product above, 5 2 {\displaystyle 5^{2}} denotes the square or second power of 5 ...
For n ≥ 2, a(n) is the prime that is finally reached when you start with n, concatenate its prime factors (A037276) and repeat until a prime is reached; a(n) = −1 if no prime is ever reached. A037274
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If none of its prime factors are repeated, it is called squarefree. (All prime numbers and 1 are squarefree.) For example, 72 = 2 3 × 3 2, all the prime factors are repeated, so 72 is a powerful number. 42 = 2 × 3 × 7, none of the prime factors are repeated, so 42 is squarefree. Euler diagram of numbers under 100:
where is the th successive prime number, and all omitted terms (a 22 to a 228) are factors with exponent equal to one (i.e. the number is ). More concisely, it is the product of seven distinct primorials: