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Mar
25
awarded  Yearling
Apr
22
awarded  Enthusiast
Apr
21
answered Are sets with similar asymptotic behavior as the primes necessarily finite additive bases?
Apr
17
comment Limit of $\frac{1}{n}\sum_{r=1}^n\frac{n\ (\mathrm{mod}\ r)}{r}$
To add to Emil's nice answer: The determination of the OP's limit is implicit in Dirichlet's estimate for the average order of the divisor function, namely $\sum_{k=1}^{n} d(k) = n \log{n} + (2\gamma-1)n + O(\sqrt{n})$. (See section 18.2 in Hardy and Wright, for example.) To see the connection, notice that the sum in question is exactly the number of ordered pairs of natural numbers $(r,s)$ with $rs \leq n$ (and now compare with Emil's answer above).
Apr
7
answered The sum of multiplicative inverses of odd numbers
Apr
5
answered Finite sums of prime numbers $\geq x$
Apr
4
comment Finite sums of prime numbers $\geq x$
Lemma 1 of Erdos's 1974 paper with Benkoski "On weird and pseudoperfect numbers" is: There is an absolute constant $c$ such that every integer $m > c p_k$ is the distinct sum of primes not less than $p_k$. (Here $p_k$ is the $k$th prime, in the usual order.) No proof is given. "The lemma is probably well known and, in any case, easily follows by Brun's method."
Mar
28
comment Binary representation of powers of 3
Alternatively, one can shorten the analysis using the following theorem of Bang: $2^n-1$ has a primitive prime divisor except for $n=1$ and $n=6$.
Mar
24
awarded  Editor
Mar
24
revised Primes $ 1 + x^2 + y^2$
added 103 characters in body
Mar
24
awarded  Teacher
Mar
24
answered Primes $ 1 + x^2 + y^2$