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visits | member for | 3 years, 5 months |
seen | Jul 6 '12 at 21:43 | |
stats | profile views | 69 |
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$ |