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The prime quadruplets are pairs of twin primes with only one odd number between them. The sum of the reciprocals of the numbers in prime quadruplets is approximately 0.8706. The sum of the reciprocals of the perfect powers (including duplicates) is 1. The sum of the reciprocals of the perfect powers (excluding duplicates) is approximately 0.8745 .
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.
One record from this search is that 3 325 581 707 333 960 528 is the smallest number that cannot be written as a sum of two primes where one is smaller than 9781. Cully-Hugill and Dudek prove a (partial and conditional) result on the Riemann hypothesis: there exists a sum of two odd primes in the interval (x, x + 9696 log^2 x] for all x ≥ 2.
Perfect number. In number theory, a perfect number is a positive integer that is equal to the sum of its positive proper divisors, that is, divisors excluding the number itself. For instance, 6 has proper divisors 1, 2 and 3, and 1 + 2 + 3 = 6, so 6 is a perfect number. The next perfect number is 28, since 1 + 2 + 4 + 7 + 14 = 28.
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.
Calculus. In mathematics, the harmonic series is the infinite series formed by summing all positive unit fractions : The first terms of the series sum to approximately , where is the natural logarithm and is the Euler–Mascheroni constant. Because the logarithm has arbitrarily large values, the harmonic series does not have a finite limit: it ...
So, 6 is a perfect number because the proper divisors of 6 are 1, 2, and 3, and 1 + 2 + 3 = 6. [2] [4] There is a one-to-one correspondence between the Mersenne primes and the even perfect numbers, but it is unknown whether there exist odd perfect numbers or not.
n even with an odd number of prime factors 1 2 9 21 100 180 1010 6067 + + ... The sum of the reciprocals of the palindromic numbers is a convergent series, whose ...