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A prime number (or prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself. By Euclid's theorem , there are an infinite number of prime numbers. Subsets of the prime numbers may be generated with various formulas for primes .
1566 = number k such that k 64 + 1 is prime; 1567 = number of partitions of 24 with at least one distinct part [200] 1568 = Achilles number [343] 1569 = 2 × 28 2 + 1 = number of different 2 × 2 determinants with integer entries from 0 to 28 [199] 1570 = 2 × 28 2 + 2 = number of points on surface of tetrahedron with edgelength 28 [141] 1571 ...
A factorial x! is the product of all numbers from 1 to x. The first: 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800, 39916800, 479001600 (sequence A000142 in the OEIS). 0! = 1 is sometimes included. A k-smooth number (for a natural number k) has its prime factors ≤ k (so it is also j-smooth for any j > k).
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Klauber's 1932 paper describes a triangle in which row n contains the numbers (n − 1) 2 + 1 through n 2. As in the Ulam spiral, quadratic polynomials generate numbers that lie in straight lines. Vertical lines correspond to numbers of the form k 2 − k + M. Vertical and diagonal lines with a high density of prime numbers are evident in the ...
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