# examples of ranges of consecutive integers for Erdős-Woods numbers

The most famous example of 16 as an Erdős-Woods number (http://planetmath.org/ErdHosWoodsNumber) is the range of 16 consecutive integers starting with 2184.

Another $n$ for $k=16$ is 2044224, which we obtained by multiplying 2184 by 936. The factorization is $2044224=2^{6}\times 3^{3}\times 7\times 13^{2}$, while $2044224+16=2044240=2^{4}\times 5\times 11\times 23\times 101$. The table of factorizations

2044225 $5^{2}\times 81769$ $2\times 1022113$ $3\times 681409$ $2^{2}\times 511057$ $11\times 19\times 9781$ $2\times 3\times 5\times 68141$ $7^{2}\times 41719$ $2^{3}\times 59\times 61\times 71$ $3^{2}\times 17\times 31\times 431$ $2\times 1009\times 1013$ $5\times 107\times 3821$ $2^{2}\times 3\times 170353$ $13\times 67\times 2347$ $2\times 7\times 151\times 967$ $3\times 29\times 23497$

shows that each of the numbers in this range shares at least one factor with one if not both of the numbers capping the range.

Next we have a slightly longer example, this one for $k=34$. The smallest matching $n$ is 47563752566, a squarefree number with a factorization of $2\times 11\times 17\times 23\times 41\times 157\times 859$. The number capping the end of the range is the decidedly non-squarefree 47563752600, with a factorization of $2^{3}\times 3^{2}\times 5^{2}\times 7\times 13\times 17\times 19\times 29% \times 31$. While the size of these numbers forbids verification on your typical pocket calculator, these numbers are well within the reach of a Javascript implementation of trial division. Here we could be tempted to omit the even numbers, as they obviously share 2 as a prime factor with the range start and the range end, as well as multiples of 3 or 5 as they thus share factors with the range end. But, on the hope that it turns out to be at least a little bit instructive, the factorizations of all the numbers in our chosen range is given.

47563752567 $3\times 3719\times 4263131$ $2^{3}\times 71\times 199\times 420799$ $31\times 163\times 9412973$ $2\times 3\times 5\times 1585458419$ $29\times 12941\times 126739$ $2^{2}\times 7^{2}\times 242672207$ $3^{2}\times 4657\times 1134821$ $2\times 13\times 823\times 991\times 2243$ $5^{2}\times 31769\times 59887$ $2^{7}\times 3\times 349\times 354911$ $11\times 397\times 593\times 18367$ $2\times 173\times 137467493$ $3\times 7\times 3257\times 695407$ $2^{2}\times 5\times 83\times 617\times 46439$ $19\times 2503355399$ $2\times 3^{4}\times 53\times 59\times 93893$ $17\times 43\times 5171\times 12583$ $2^{3}\times 149\times 39902477$ $3\times 5\times 67\times 47327117$ $2\times 7\times 3397410899$ $13^{2}\times 281442323$ $2^{2}\times 3\times 11\times 4513\times 79843$ $23\times 61\times 151\times 224513$ $2\times 5\times 4756375259$ $3^{2}\times 5284861399$ $2^{4}\times 47\times 63249671$ $7\times 6794821799$ $2\times 3\times 7927292099$ $5\times 32503\times 292673$ $2^{2}\times 11890938149$ $3\times 15854584199$ $2\times 23781876299$ $11\times 37\times 127\times 373\times 2467$
Title examples of ranges of consecutive integers for Erdős-Woods numbers ExamplesOfRangesOfConsecutiveIntegersForErdHosWoodsNumbers 2013-03-22 17:38:16 2013-03-22 17:38:16 PrimeFan (13766) PrimeFan (13766) 5 PrimeFan (13766) Example msc 11A05