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Name First elements Short description OEIS Mersenne prime exponents : 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, ... Primes p such that 2 p − 1 is prime.: A000043 ...
A Markov number or Markoff number is a positive integer x, y or z that is part of a solution to the Markov Diophantine equation + + =, studied by Andrey Markoff (1879, 1880). The first few Markov numbers are 1, 2, 5, 13, 29, 34, 89, 169, 194, 233, 433, 610, 985, 1325, ...
Lucas numbers have L 1 = 1, L 2 = 3, and L n = L n−1 + L n−2. Primefree sequences use the Fibonacci recursion with other starting points to generate sequences in which all numbers are composite. Letting a number be a linear function (other than the sum) of the 2 preceding numbers. The Pell numbers have P n = 2P n−1 + P n−2.
The first examples of such a p, for which π (p) is smaller than 2(p+1), are π (47) = 2(47 + 1)/3 = 32, π (107) = 2(107 + 1)/3 = 72 and π (113) = 2(113 + 1)/3 = 76. ( See the table below ) It follows from above results, that if n = p k is an odd prime power such that π ( n ) > n , then π ( n )/4 is an integer that is not greater than n .
In words: the first two numbers in the sequence are both 2, and each successive number is formed by adding twice the previous Pell–Lucas number to the Pell–Lucas number before that, or equivalently, by adding the next Pell number to the previous Pell number: thus, 82 is the companion to 29, and 82 = 2 × 34 + 14 = 70 + 12.
The list on the right shows the Fibonacci sequence: 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377. The 2, 8, and 9 resemble Arabic numerals more than Eastern Arabic numerals or Indian numerals . Leonardo Fibonacci was a Pisan mathematician who had studied in the Pisan trading colony of Bugia , in what is now Algeria , [ 15 ] and he ...
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4 4 4 = 4 ↑↑ 3 = 2 512 ≈ 1.34 × 10 154 ≈ ... Class 1 – numbers between six and 1,000,000=10 6 – is defined to contain numbers whose decimal expressions ...