User tdnoe - MathOverflow 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 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 Answer by tdnoe for Intervals with large numbers of primes 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 What is the name for(a^2 + b^2 + c^2 +...)/(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 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 Answer by tdnoe for What is the lower bound for highly composite numbers? 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 Answer by tdnoe for Factoring blocks of numbers 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 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 Answer by tdnoe for Does there exist a pure recurrence formula with polynomial coefficients for Fibonacci(2^n)? 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 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 Answer by tdnoe for Irreducible polynomials with constrained coefficients 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 Comment by tdnoe 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 Comment by tdnoe 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 Comment by tdnoe 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 Comment by tdnoe 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 Comment by tdnoe 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 Comment by tdnoe 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 Comment by tdnoe 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 Comment by tdnoe 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 Comment by tdnoe 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 Comment by tdnoe 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 Comment by tdnoe 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 Comment by tdnoe 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 Comment by tdnoe 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 Comment by tdnoe 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 Comment by tdnoe 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).