# Integer Sequences

## Combinatorics

Sequence Database. Numberphile. What's Special About This Number? Prime generating fractions. I posted a couple prime-generating fractions on Google+ this weekend and said that I’d post an explanation later.

Here it goes. The decimal expansion of 18966017 / 997002999 is where each group of three digits is a prime number. The fraction 4099200041 / 999700029999 produces primes in groups of four digits: What’s going on here? Euler discovered in 1772 that n2 – n + 41 is prime for values of n from -40 to 40. Numbers Aplenty. AC Applied Combinatorics. Skip to main content. Contributing a b-file or other file to the On-Line Encyclopedia of Integer Sequences. Contributing a b-file or other file to the On-Line Encyclopedia of Integer Sequences About b-files Entries in the The On-Line Encyclopedia of Integer Sequences give at most three lines of terms (about 200 characters - a carry-over from the days when the database was on punched cards).

Often one wants more terms than this, so many sequences now have "b-files", giving 10000 or more terms. The format for a b-file is: 1 2 2 3 3 5 4 7 5 11 6 13 7 17 8 19 9 23 10 29 etc. with one line per entry. This is a list of pairs "n a(n)", where n starts at the correct offset for the sequence and a(n) is the n-th term. n should begin at the left then a single space then a(n). When you edit a sequence, in the Links section you will see a place where you can upload a b-file, or other file that is to be associated with this sequence.

Fermat numbers are numbers of the form Fermat numbers: (Cf. A000215) Fermat numbers in base 2 representation. Formulae Recurrences. Jj's useful and ugly FXT page. Here you find the FXT library the fxtbook: "Matters Computational" (was: "Algorithms for Programmers") the amorphous FFT bucket moved to the fftpage The FXT library Download the latest FXT version: fxt-2018.07.03.tar.gz (approx. 1800kB), distributed under the GPL.

FXT is a library of low-level algorithms. Its main focus is on bit-manipulations, combinatorial generation, and fast transforms. The library is accompanied by the fxtbook. The complete list of demos contains about 500 programs, listed by topic. Here is a short description, the Linux Software Map (LSM) file fxt.lsm. Great Internet Mersenne Prime Search - PrimeNet. Get Elements of Lists—Wolfram Language Documentation.

Publications - Value of listing OEIS contributions on an academic CV - Academia Stack Exchange. The Online Encyclopedia of Integer Sequences (OEIS) is a large online database of number sequences.

It's popular among professional and amateur mathematicians, and its entries are widely cited. Like Wikipedia, most of its content is authored and edited by the community; no particular qualifications are required to participate, though in my experience most active editors are working mathematicians or scientists. I've occasionally seen academic CVs which list OEIS contributions among the publications (usually in a separate subsection). I've also seen CVs of known contributors which make no mention whatsoever of their OEIS contributions.

I was wondering, then, whether listing OEIS contributions is generally a good idea when applying for academic jobs such as post-docs and professorships. Formula di Perron. Da Wikipedia, l'enciclopedia libera.

In teoria analitica dei numeri, la formula di Perron è una formula che permette di calcolare la somma di una funzione aritmetica tramite una trasformata di Mellin inversa. La formula prende il nome da Oskar Perron. Enunciato[modifica | modifica wikitesto] Sia una funzione aritmetica, e sia la sua serie di Dirichlet corrispondente. Infinitude of Primes Via Powers of 2. The following statement directly implies infinitude of primes: For a positive integer the expression has at least distinct prime factors.

Proof The proof is by induction and employs the following identity Note that, setting, The two factors of are mutually prime, since each is an odd number while their difference equals and, so, the only divisors it has are the powers of Now, a number with exactly factors. Assuming the statement holds for an integer i.e., that has at least distinct prime factors, observe that, according to (2), with the second factor having divisors different from those of which, by the inductive hypothesis has at least distinct prime divisors. Remark It was shown by Richard Green from the Unversity of Colorado that the above proof, although simple, can still be made more transparent. The induction then follows as before.

References A. [an error occurred while processing this directive] |Contact||Front page||Contents||Algebra||Up| Surveyors formula for area of a polygon. If you know the vertices of a polygon, how do you compute its area?

This seems like this could be a complicated, with special cases for whether the polygon is convex or maybe other considerations. But as long as the polygon is “simple,” i.e. the sides meet at vertices but otherwise do not intersect each other, then there is a general formula for the area.