most recent 30 from http://mathoverflow.net 2013-05-19T04:23:15Z http://mathoverflow.net/feeds/user/2360 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/62444/are-there-ever-three-perfect-powers-between-consecutive-squares tdnoe 2011-04-20T18:20:41Z 2012-12-11T05:04:00Z

<p>I have not seen this question treated in the literature. Does anyone have more information? There are several OEIS sequences (<a href="http://oeis.org/A097056" rel="nofollow">A097056</a>, <a href="http://oeis.org/A117896" rel="nofollow">A117896</a>, <a href="http://oeis.org/A117934" rel="nofollow">A117934</a>) dealing with this question, but no answers.</p>

http://mathoverflow.net/questions/26032/intervals-with-large-numbers-of-primes/26052#26052 tdnoe 2010-05-26T19:10:28Z 2011-11-16T16:50:37Z

<p>OEIS sequence <a href="http://oeis.org/A120934" rel="nofollow">A120934</a> gives the least prime $p$ such that the interval $[p,p+\log(p)]$ contains $n$ primes.</p>

http://mathoverflow.net/questions/47738/what-is-the-name-fora2-b2-c2-a-b-c tdnoe 2010-11-30T00:24:55Z 2010-11-30T11:08:41Z

<p>That is, the sum of squares of some numbers divided by the sum of the numbers. The term "anti-harmonic mean" has been coined for this quantity. I'm hoping there is a better name.</p>

http://mathoverflow.net/questions/44865/are-the-largely-composite-numbers-the-same-as-the-fully-composite-numbers tdnoe 2010-11-04T19:37:26Z 2010-11-07T18:22:23Z

<p>Let $d(n)$ be the number of divisors of $n$. Let $p(n)$ be the product of the divisors of $n$. Ramanujan called a number $n$ largely composite if $d(n) \ge d(m)$ for $m &lt; n$. Let's call $n$ fully composite if $p(n) \ge p(m)$ for $m &lt; n$. It is conjectured that largely composite numbers (LCN) are the same as fully composite numbers (FCN). It is easy to show that an LCN is an FCN. Is every FCN an LCN? In the OEIS, these are sequences <a href="http://oeis.org/classic/A067128" rel="nofollow">http://oeis.org/classic/A067128</a> and <a href="http://oeis.org/classic/A034287" rel="nofollow">http://oeis.org/classic/A034287</a>. These two sequences are the same for the 105834 terms less than 10^150.</p> <p>This question is interesting because it connects the number of divisors to the product of divisors.</p>

http://mathoverflow.net/questions/43103/what-is-the-lower-bound-for-highly-composite-numbers/44894#44894 tdnoe 2010-11-04T23:27:02Z 2010-11-05T16:57:20Z

<p>It would be nice to have an inequality $n \ge f(x)$. If the poser wants numerical results, here are two:</p> <p>The least number having exactly x divisors is given by OEIS sequence <a href="http://www.oeis.org/classic/A005179" rel="nofollow">http://www.oeis.org/classic/A005179</a>. It is a pretty wild function. The nice paper by Grost is recommended.</p> <p>The least number having x (or more) divisors is given by the OEIS sequence <a href="http://www.oeis.org/classic/A061799" rel="nofollow">http://www.oeis.org/classic/A061799</a>.</p>

http://mathoverflow.net/questions/41725/factoring-blocks-of-numbers/41831#41831 tdnoe 2010-10-11T20:30:51Z 2010-10-11T20:30:51Z

<p>Using a sieve of Eratosthenes approach, it is easy to create a list $S$ of the smallest prime factor of every number less than $N$. Then to factor any $n &lt; N$, we just recursively look up the factors: let $n_1=n$, $f_i = S(n_i)$, and $n_{i+1} = n_i/f_i$. With 4 GB of memory, numbers less than ${10}^9$ are quickly factored. Although this algorithm takes more memory, it is much faster than trial division.</p>

http://mathoverflow.net/questions/41802/a-hierarchy-of-k-highly-composite-numbers tdnoe 2010-10-11T16:13:49Z 2010-10-11T19:05:17Z

<p>Let $\sigma_k(n)$ denote the sum of the k-th powers of the divisors of n. For any real value of k, we can find a sequence of numbers $s_k$ that has increasing values of n at which $\sigma_k(n)$ attains a new maximum. For k=0, that sequence is called the highly composite numbers, which is <a href="http://oeis.org/classic/A002182" rel="nofollow">A002182</a> in the OEIS database. From numerical experimentation, it appears that if i > j >= 0, then $s_j$ is a subsequence of $s_i$. Is this a known result? If not, any ideas on how to prove it?</p>

http://mathoverflow.net/questions/32771/does-there-exist-a-pure-recurrence-formula-with-polynomial-coefficients-for-fibon/32829#32829 tdnoe 2010-07-21T18:31:24Z 2010-07-21T18:31:24Z

<p>Interestingly, the recursion $u_{n+1} = (u_n + 5/u_n)/2$, with $u_0=1$, gives the fractions $Lucas(2^n)/Fibonacci(2^n)$.</p>

http://mathoverflow.net/questions/19146/do-six-consecutive-numbers-always-contain-an-abundant-or-perfect-number tdnoe 2010-03-23T21:15:34Z 2010-03-24T00:10:24Z

<p>Let sigma(n) be the sum of the divisors of n. Take six consecutive numbers. It appears that at least one of the six has sigma(n) >= 2n. Has this been proved?</p>

http://mathoverflow.net/questions/7969/irreducible-polynomials-with-constrained-coefficients/8086#8086 tdnoe 2009-12-07T07:34:52Z 2009-12-07T07:34:52Z

<p>Sequence <a href="http://www.research.att.com/~njas/sequences/A087481" rel="nofollow">http://www.research.att.com/~njas/sequences/A087481</a> lists the number of such irreducible polynomials up to degree 18.</p>

http://mathoverflow.net/questions/47738/what-is-the-name-fora2-b2-c2-a-b-c tdnoe 2010-11-30T00:46:02Z 2010-11-30T00:46:02Z

Yes, I do. At least it has some history. Thanks!

http://mathoverflow.net/questions/46068/how-small-can-a-sum-of-a-few-roots-of-unity-be/46069#46069 tdnoe 2010-11-16T18:35:48Z 2010-11-16T18:35:48Z

The OEIS sequence <a href="http://oeis.org/A108380" rel="nofollow">oeis.org/A108380</a> gives the least number of distinct n-th roots of unity summing to the smallest possible nonzero magnitude. There is also a plot of the least magnitude for n up to 81.

http://mathoverflow.net/questions/44865/are-the-largely-composite-numbers-the-same-as-the-fully-composite-numbers tdnoe 2010-11-04T21:53:49Z 2010-11-04T21:53:49Z

It appears to be something like c (log x)^2.

http://mathoverflow.net/questions/44844/galois-groups-of-a-family-of-polynomials tdnoe 2010-11-04T20:28:48Z 2010-11-04T20:28:48Z

For 241, the discriminant is 2^240 11^478 241^238 -- clearly a square.

http://mathoverflow.net/questions/44865/are-the-largely-composite-numbers-the-same-as-the-fully-composite-numbers tdnoe 2010-11-04T20:10:27Z 2010-11-04T20:10:27Z

Actually, it can be proved that $p(n) \ne p(m)$ for $n \ne m$. Using the identity $p(n) = n^{(d(n)/2)}$, it is easy to show that $d(n) \ge d(m)$ implies $p(n) \ge p(m)$.

http://mathoverflow.net/questions/44844/galois-groups-of-a-family-of-polynomials tdnoe 2010-11-04T19:50:15Z 2010-11-04T19:50:15Z

Your three exceptions 7, 17, 97 are the first three primes of the form 2*q^2-1 where q is prime, which is sequence A092057. Is the next exception 241?

http://mathoverflow.net/questions/42897/how-many-solutions-are-there-to-abcd-mod-p-where-p-is-a-prime-and-1a-b-c-dp tdnoe 2010-10-20T22:05:52Z 2010-10-20T22:05:52Z

See OEIS sequence <a href="http://www.oeis.org/A020478" rel="nofollow">oeis.org/A020478</a> for the general case of p being a positive integer

http://mathoverflow.net/questions/41802/a-hierarchy-of-k-highly-composite-numbers tdnoe 2010-10-19T15:56:59Z 2010-10-19T15:56:59Z

There are ten counterexamples in the new sequence oeis.org/classic/A181309

http://mathoverflow.net/questions/41802/a-hierarchy-of-k-highly-composite-numbers tdnoe 2010-10-11T17:32:18Z 2010-10-11T17:32:18Z

It was just pointed out to me that 1084045767585249647898720000 provides a counterexample: it is in $s_0$ but not in $s_1$.

http://mathoverflow.net/questions/39373/does-nm-t-have-infinitely-many-solutions-besides-trivial-ones/39374#39374 tdnoe 2010-09-20T19:54:26Z 2010-09-20T19:54:26Z

In the 1975 Erdos paper mentioned in OEIS sequence A003135, this is considered a trivial solution. See pages 27-28 of that paper.

http://mathoverflow.net/questions/39373/does-nm-t-have-infinitely-many-solutions-besides-trivial-ones tdnoe 2010-09-20T18:21:10Z 2010-09-20T18:21:10Z

The OEIS sequence <a href="http://oeis.org/classic/A003135" rel="nofollow">oeis.org/classic/A003135</a> is related to this question.

http://mathoverflow.net/questions/34669/is-there-any-progress-toward-solving-gilbreaths-conjecture/34691#34691 tdnoe 2010-08-06T05:22:44Z 2010-08-06T05:22:44Z

The question of how many increasing sequences of integers have the Gilbreath property is answered in <a href="http://oeis.org/classic/A080839" rel="nofollow">oeis.org/classic/A080839</a>. It doesn't make the primes seem that special.

http://mathoverflow.net/questions/34474/conjecture-about-prime-gaps tdnoe 2010-08-05T16:07:12Z 2010-08-05T16:07:12Z

The terms of -h[2n] are in sequence <a href="http://oeis.org/classic/A121573" rel="nofollow">oeis.org/classic/A121573</a>, prime-gap race; difference of the cumulative sums of gaps above and below prime(2n).

http://mathoverflow.net/questions/33411/a-generalisation-of-the-equation-n-ab-ac-bc tdnoe 2010-07-26T22:32:25Z 2010-07-26T22:32:25Z

If we allow equality, the $n_0$ have been conjectured in <a href="http://oeis.org/classic/A027565" rel="nofollow">oeis.org/classic/A027565</a>, which appears to be growing exponentially.

http://mathoverflow.net/questions/32771/does-there-exist-a-pure-recurrence-formula-with-polynomial-coefficients-for-fibon/32776#32776 tdnoe 2010-07-22T15:58:43Z 2010-07-22T15:58:43Z

It there a typo here? Starting with $x_0=1$ and $x_1=1$, the recursion produces $x_2=3$ (correct), $x_3=21$ (correct), and $x_4=2016$ (should be 987).