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Let p_n be the nth prime, then the primorial (which is the analog of the usual factorial for prime numbers) is defined by p_n#=product_(k=1)^np_k. (1) The values of p_n# for ...

The totient function phi(n), also called Euler's totient function, is defined as the number of positive integers <=n that are relatively prime to (i.e., do not contain any ...

There are two definitions of the Carmichael function. One is the reduced totient function (also called the least universal exponent function), defined as the smallest integer ...

The Smarandache function mu(n) is the function first considered by Lucas (1883), Neuberg (1887), and Kempner (1918) and subsequently rediscovered by Smarandache (1980) that ...

There exist a variety of formulas for either producing the nth prime as a function of n or taking on only prime values. However, all such formulas require either extremely ...

There are two definitions of Bernoulli polynomials in use. The nth Bernoulli polynomial is denoted here by B_n(x) (Abramowitz and Stegun 1972), and the archaic form of the ...

Primorial primes are primes of the form p_n#+/-1, where p_n# is the primorial of p_n. A coordinated search for such primes is being conducted on PrimeGrid. p_n#-1 is prime ...

The important binomial theorem states that sum_(k=0)^n(n; k)r^k=(1+r)^n. (1) Consider sums of powers of binomial coefficients a_n^((r)) = sum_(k=0)^(n)(n; k)^r (2) = ...

Perfect numbers are positive integers n such that n=s(n), (1) where s(n) is the restricted divisor function (i.e., the sum of proper divisors of n), or equivalently ...

Legendre showed that there is no rational algebraic function which always gives primes. In 1752, Goldbach showed that no polynomial with integer coefficients can give a prime ...

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