Consecutive integers with the same number of factors

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The numbers 2 and 3 are both prime, so both have two factors, themselves and 1.

2 3
1 1

While there are no other examples of consecutive primes, it’s not difficult to find more pairs of consecutive integers with the same number of factors:

14 15
7 5
2 3
1 1
    

21 22
7 11
3 2
1 1

Eventually you get to sets of three consecutive integers with the same number of factors:

33 34 35
11 17 7
3 2 5
1 1 1
    

85 86 87
17 43 29
5 2 3
1 1 1

It isn’t possible to have a sequence of more than three integers with only four factors, but so far I have found this set of four consecutive integers with six factors:

242 243 244 245
121 81 122 49
22 27 61 35
11 9 4 7
2 3 2 5
1 1 1 1

I’m sure that somewhere out there I can find out whether or not a) there are longer sequences of consecutive integers which the same number of factors and b) whether there is a natural limit to the process. That’s as far as I can take it in the sleepless hours of the night though.

One thought on “Consecutive integers with the same number of factors”

  1. I suspect the answer to all these sorts of questions depends on who I am with, the formality of the speech, how much I’ve had to drink and many other factors.

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