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20 numbers and 8 primes (Posted on 2006-10-12) Difficulty: 4 of 5
I am looking for n consecutive integers such that (i) every number in the sequence is divisible by a prime <=n and (ii) every prime number <=n is a factor of at least two of the numbers. For example, consider n=3:
a) There are two primes less than or equal to 3. They are 2 and 3.
b) 6 7 8 does not work, in part because 7 is not evenly divisible by either 2 or 3
c) 8 9 10 does not work, even though all are divisible by 2 or 3, because 3 divides only one of them

There is some reason to believe that no sequence of positive integers works for n < 20. For n = 20:

1) What is the first sequence of 20 consecutive positive integers that works?
2) What is the second?
3) How often do they repeat after that?
4) What interesting number results if you add the first integer from one of the first two sequences to the last integer of the other?

By the way, this problem grew out of JLo's innocent perplexus problem "Six numbers and a prime"

See The Solution Submitted by Steve Herman    
Rating: 4.5000 (2 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
re(3): Confessions and a new problem, thanks to Charlie | Comment 5 of 7 |
(In reply to re(2): Confessions and a new problem, thanks to Charlie by brianjn)

The at sign is used in UBASIC to represent the MOD function.  When I pasted the program listing into this comment window, it saw the at sign and made a link out of it.  It doesn't affect anything; it's not really a link, and the text shows up correctly other than being blue.
  Posted by Charlie on 2006-10-13 11:46:21

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