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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.
Every integer greater than 20161 can be written as the sum of two abundant numbers. The largest even number that is not the sum of two abundant numbers is 46. [5] An abundant number which is not a semiperfect number is called a weird number. [6] An abundant number with abundance 1 is called a quasiperfect number, although none have yet been found.
Apart from 1 + 2 = 3 any subsequent Ulam number cannot be the sum of its two prior consecutive Ulam numbers. Proof: Assume that for n > 2, U n−1 + U n = U n+1 is the required sum in only one way; then so does U n−2 + U n produce a sum in only one way, and it falls between U n and U n+1.
The summation of an explicit sequence is denoted as a succession of additions. For example, summation of [1, 2, 4, 2] is denoted 1 + 2 + 4 + 2, and results in 9, that is, 1 + 2 + 4 + 2 = 9. Because addition is associative and commutative, there is no need for parentheses, and the result is the same irrespective of the order of the summands ...
Express each term of the final sequence y 0, y 1, y 2, ... as the sum of up to two terms of these intermediate sequences: y 0 = x 0, y 1 = z 0, y 2 = z 0 + x 2, y 3 = w 1, etc. After the first value, each successive number y i is either copied from a position half as far through the w sequence, or is the previous value added to one value in the ...
In the string representing the integer x, the n th digit is 1 if F −n appears in the sum that represents x; that digit is 0 otherwise. For example, 24 may be represented by the string 100101001, which has the digit 1 in places 9, 6, 4, and 1, because 24 = F −1 + F −4 + F −6 + F −9 .
In 1975, Hugh Lowell Montgomery and Bob Vaughan showed that "most" even numbers are expressible as the sum of two primes. More precisely, they showed that there exist positive constants c and C such that for all sufficiently large numbers N, every even number less than N is the sum of two primes, with at most CN 1 − c exceptions.
For example, for the array of values [−2, 1, −3, 4, −1, 2, 1, −5, 4], the contiguous subarray with the largest sum is [4, −1, 2, 1], with sum 6. Some properties of this problem are: If the array contains all non-negative numbers, then the problem is trivial; a maximum subarray is the entire array.