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	$5^2\times 81769$
2044226	$2\times 1022113$
2044227	$3\times 681409$
2044228	$2^2\times 511057$
2044229	$11\times 19\times 9781$
2044230	$2\times 3\times 5\times 68141$
2044231	$7^2\times 41719$
2044232	$2^3\times 59\times 61\times 71$
2044233	$3^2\times 17\times 31\times 431$
2044234	$2\times 1009\times 1013$
2044235	$5\times 107\times 3821$
2044236	$2^2\times 3\times 170353$
2044237	$13\times 67\times 2347$
2044238	$2\times 7\times 151\times 967$
2044239	$3\times 29\times 23497$

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	$3\times 3719\times 4263131$
47563752568	$2^3\times 71\times 199\times 420799$
47563752569	$31\times 163\times 9412973$
47563752570	$2\times 3\times 5\times 1585458419$
47563752571	$29\times 12941\times 126739$
47563752572	$2^2\times 7^2\times 242672207$
47563752573	$3^2\times 4657\times 1134821$
47563752574	$2\times 13\times 823\times 991\times 2243$
47563752575	$5^2\times 31769\times 59887$
47563752576	$2^7\times 3\times 349\times 354911$
47563752577	$11\times 397\times 593\times 18367$
47563752578	$2\times 173\times 137467493$
47563752579	$3\times 7\times 3257\times 695407$
47563752580	$2^2\times 5\times 83\times 617\times 46439$
47563752581	$19\times 2503355399$
47563752582	$2\times 3^4\times 53\times 59\times 93893$
47563752583	$17\times 43\times 5171\times 12583$
47563752584	$2^3\times 149\times 39902477$
47563752585	$3\times 5\times 67\times 47327117$
47563752586	$2\times 7\times 3397410899$
47563752587	${13}^2\times 281442323$
47563752588	$2^2\times 3\times 11\times 4513\times 79843$
47563752589	$23\times 61\times 151\times 224513$
47563752590	$2\times 5\times 4756375259$
47563752591	$3^2\times 5284861399$
47563752592	$2^4\times 47\times 63249671$
47563752593	$7\times 6794821799$
47563752594	$2\times 3\times 7927292099$
47563752595	$5\times 32503\times 292673$
47563752596	$2^2\times 11890938149$
47563752597	$3\times 15854584199$
47563752598	$2\times 23781876299$
47563752599	$11\times 37\times 127\times 373\times 2467$

Title	examples of ranges of consecutive integers for Erdős-Woods numbers
Canonical name	ExamplesOfRangesOfConsecutiveIntegersForErdHosWoodsNumbers
Date of creation	2013-03-22 17:38:16
Last modified on	2013-03-22 17:38:16
Owner	PrimeFan (13766)
Last modified by	PrimeFan (13766)
Numerical id	5
Author	PrimeFan (13766)
Entry type	Example
Classification	msc 11A05