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where f (2k−1) is the (2k − 1)th derivative of f and B 2k is the (2k)th Bernoulli number: B 2 = 1 / 6 , B 4 = − + 1 / 30 , and so on. Setting f ( x ) = x , the first derivative of f is 1, and every other term vanishes, so [ 15 ]
[1] [2] For example, 20 is a primitive abundant number because: The sum of its proper divisors is 1 + 2 + 4 + 5 + 10 = 22, so 20 is an abundant number. The sums of the proper divisors of 1, 2, 4, 5 and 10 are 0, 1, 3, 1 and 8 respectively, so each of these numbers is a deficient number. The first few primitive abundant numbers are:
The series' terms (1, −2, 3, −4, ...) do not approach 0; therefore 1 − 2 + 3 − 4 + ... diverges by the term test.Divergence can also be shown directly from ...
Contrast the term primitive notion, which is a core concept not defined in terms of other concepts. Primitive notions are used as building blocks to define other concepts. Contrast also the term undefined behavior in computer science, in which the term indicates that a function may produce or return any result, which may or may not be correct.
The only odd practical number is 1, because if is an odd number greater than 2, then 2 cannot be expressed as the sum of distinct divisors of . More strongly, Srinivasan (1948) observes that other than 1 and 2, every practical number is divisible by 4 or 6 (or both).
The aliquot sequence starting with a positive integer k can be defined formally in terms of the sum-of-divisors function σ 1 or the aliquot sum function s in the following way: [1] = = = > = = = If the s n-1 = 0 condition is added, then the terms after 0 are all 0, and all aliquot sequences would be infinite, and we can conjecture that all aliquot sequences are convergent, the limit of these ...
"subtract if possible, otherwise add": a(0) = 0; for n > 0, a(n) = a(n − 1) − n if that number is positive and not already in the sequence, otherwise a(n) = a(n − 1) + n, whether or not that number is already in the sequence.
Both 2 and 3 are primitive λ-roots modulo 5 and also primitive roots modulo 5. n = 8. The set of numbers less than and coprime to 8 is {1,3,5,7} . Hence φ(8) = 4 and λ(8) must be a divisor of 4. In fact λ(8) = 2 since ().