what are derative numbers

Innumber theory, thearithmetic derivative, ornumber derivative, is a function defined forintegers, based onprime factorization, by analogy with theproduct rulefor thederivative of a functionthat is used inmathematical analysis.

Fornatural numbersthe arithmetic derivative is defined as follows:

  • for any prime.
  • for any(Leibniz rule).

To coincide with the Leibniz ruleis defined to be, as is. Explicitly, assume that

whereare distinct primes andare positive integers. Then

The arithmetic derivative also preserves the power rule (for primes):

whereis prime andis a positive integer. For example,

The sequence of number derivatives fork= 0, 1, 2, ... begins (sequenceA003415inOEIS):

0, 0, 1, 1, 4, 1, 5, 1, 12, 6, 7, 1, 16, 1, 9, ....

E. J. Barbeauwas the first to formalize this definition. He extended it to all integers by proving thatuniquely defines the derivative over the integers. Barbeau also further extended it to rational numbers,showing that the familiarquotient rulegives a well-defined derivative onQ:

Victor UfnarovskiandBo Åhlanderexpanded it to certain irrationals. In these extensions, the formula above still applies, but the exponentsare allowed to be arbitrary rational numbers.

Thelogarithmic derivativeis atotally additive function.

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