In this context, sigma() means the sum-of-divisors function. The formula for phi(N) is N * product (1 - 1/P) product taken over primes P that divide N. The formula for sigma(N) is N * product (1 + 1/P + 1/P^2 + ... 1/P^E) with P dividing N and P^E being the highest power of P that divides N. If N = P*Q, with P & Q distinct primes, phi(N) = (P-1)(Q-1) = N - P - Q + 1, and sigma(N) = (P+1)(Q+1) = N + P + Q + 1. If N = P^2 * Q, with P & Q distinct primes, phi(N) = (P^2 - P) (Q-1), while sigma(N) = (P^2 + P + 1) (Q+1). When N>2, phi(N) is even. sigma(N) is even except when N is a square or twice a square. If A and B have no common factor, both phi and sigma are multiplicative: phi(AB) = phi(A)phi(B) and ditto for sigma. One consequence is that both phi(N) and sigma(N) can be computed by applying them to the prime-power factorization of N. Rich ------- Quoting Dan Asimov <asimov@msri.org>:
What is sigma(n) ?
?Dan
On Nov 20, 2015, at 5:05 PM, David Wilson <davidwwilson@comcast.net> wrote:
Is
1. Is phi(n) + sigma(n) - 2n >= 0 for all n > 1?
2. Is phi(n) + sigma(n) - 2n = 5 for any n?
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