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A cluster prime is a prime p such that every even natural number k ≤ p − 3 is the difference of two primes not exceeding p. 3, 5, 7, 11, 13, 17, 19, 23, ... (OEIS: A038134) All odd primes between 3 and 89, inclusive, are cluster primes. The first 10 primes that are not cluster primes are: 2, 97, 127, 149, 191, 211, 223, 227, 229, 251.
A sphenic number has Ω(n) = 3 and is square-free (so it is the product of 3 distinct primes). The first: 30, 42, 66, 70, 78, 102, 105, 110, 114, 130, 138, 154 (sequence A007304 in the OEIS). a 0 (n) is the sum of primes dividing n, counted with multiplicity. It is an additive function.
[3] Perfect numbers are natural numbers that equal the sum of their positive proper divisors, which are divisors excluding the number itself. So, 6 is a perfect number because the proper divisors of 6 are 1, 2, and 3, and 1 + 2 + 3 = 6. [2] [4]
The sum of the first odd integers, beginning with one, is a perfect square: 1, 1 + 3, 1 + 3 + 5, 1 + 3 + 5 + 7, etc. This explains Galileo's law of odd numbers : if a body falling from rest covers one unit of distance in the first arbitrary time interval, it covers 3, 5, 7, etc., units of distance in subsequent time intervals of the same length.
A list of articles about numbers (not about numerals). Topics include powers of ten, notable integers, prime and cardinal numbers, and the myriad system.
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Pierre de Fermat gave a criterion for numbers of the form 8a + 1 and 8a + 3 to be sums of a square plus twice another square, but did not provide a proof. [1] N. Beguelin noticed in 1774 [2] that every positive integer which is neither of the form 8n + 7, nor of the form 4n, is the sum of three squares, but did not provide a satisfactory proof. [3]
T(n) is the sum of the first n triangular numbers, with T(0) = 0 (empty sum). A000292: Square pyramidal numbers: 0, 1, 5, 14, 30, 55, 91, 140, 204, 285, ... n(n + 1)(2n + 1) / 6 : The number of stacked spheres in a pyramid with a square base. A000330: Cube numbers n 3: 0, 1, 8, 27, 64, 125, 216, 343, 512, 729, ... n 3 = n × n × n ...
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