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Apr
27 |
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Bounds for the largest divisor of n less than n^0.5
The state of the art for questions such as this is Kevin Ford's paper arxiv.org/pdf/math/0401223v5.pdf. For the $f(n)$ given in the question, or in fact any $f(n)$ with $f(n) = n^{1/2+o(1)}$, one gets from Ford's work that the set of $n$ with $d(n) \ge f(n)$ has asymptotic density zero. |

Feb
5 |
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Is Euler's totient sub-homogeneous of degree 1?
Another way of seeing this: $\phi(mn)$ is the number of integers in $[1,mn]$ coprime to $mn$, while $m\phi(n)$ is the number of integers in $[1,mn]$ coprime to $n$. |

Jan
1 |
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3-7 primes in base 10
I will add that ``almost-prime'' results of this sort have been obtained by Dartyge and Mauduit: Almost primes whose expansion in base $r$ misses some digits. (Nombres presque premiers dont l'écriture en base $r$ ne comporte pas certains chiffres.) (French) J. Number Theory 81, No.2, 270-291 (2000) |

Jan
1 |
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3-7 primes in base 10
Some questions of a similar flavor, for the much denser set of squarefree numbers replacing the set of primes, are considered by Filaseta and Konyagin in this Acta Arithmetica paper: matwbn.icm.edu.pl/ksiazki/aa/aa74/aa7431.pdf In particular, they prove that for each $2 \leq b \leq 5$, there are infinitely many squarefree numbers in base $b$ consisting only of the digits $0$ and $1$. When $b=10$, they can show there are infinitely many squarefree numbers consisting just of the digits $0$, $1$, and $2$. |

Dec
5 |
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Average orders of multiplicative functions
Chapter IV of this book is a compendium of results related to four non-multiplicative functions (the largest prime divisor, the smallest prime divisor, and the sum of the prime divisors, counted with or without multiplicity). Now there may still be a result in this chapter that fits the bill, but maybe a more precise reference could be given? |

Nov
16 |
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Least prime primitive root
It looks from Wang's paper (which can be found by searching Google books for his collected works) that he actually shows the partial sums of $\Lambda(n) e^{-n/x}$, taken over primitive roots $n$ up to about $(log p)^{C}$, is positive. So this gives a prime small power $n$ which is a primitive root mod $p$. But then the prime which $n$ is a power of must also be a primitive root mod $p$. |

Nov
16 |
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Least prime primitive root
OK, answer posted! |