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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).
Terms with infinitely many sine factors would necessarily be equal to zero. When only finitely many of the angles are nonzero then only finitely many of the terms on the right side are nonzero because all but finitely many sine factors vanish. Furthermore, in each term all but finitely many of the cosine factors are unity.
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.
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:
90 is the only number to have an aliquot sum of 144 = 12 2. Only three numbers have a set of divisors that generate a sum equal to 90, they are 40, 58, and 89. [3] 90 is also the twentieth abundant [4] and highly abundant [5] number (with 20 the first primitive abundant number and 70 the second). [6] The number of divisors of 90 is 12. [7]
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Given an integer of unknown form, these methods are usually applied before general-purpose methods to remove small factors. [10] For example, naive trial division is a Category 1 algorithm. Trial division; Wheel factorization; Pollard's rho algorithm, which has two common flavors to identify group cycles: one by Floyd and one by Brent.