(adsbygoogle = window.adsbygoogle || []).push({}); Other "primes"

If, instead a defining a prime as an integer divisable only by 1 or itself, we define it as an integer not divisable by a number other than 1, 2, half of itself, or itself, (numbers not being divisable by 2 still "able" to be primes) then we generate a new "prime" sequence:

1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 13, 14, 17, 19, 22, 23, 26, 27 .... etc

This way we allow even numbers to be "prime". We could also define a prime as a number not divisable by any number other than 1, 2, 3, third of self, half of self, self. As we do so we are clearly increasing the number of "primes". The question then is, are any of these sequences any easier to predict than the traditional primes?

If we let [tex]primes_{1}[/tex] be the traditional primes, then:

[tex]primes_{1} = 1, 3, 5, 7, 11, 13, 17, 19, 23, 27 ...[/tex]

[tex]primes_{2} = 1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 13, 14, 17, 19, 22, 23, 26, 27 ...[/tex]

[tex]primes_{3} = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 19, 21, 22, 23, 26, 27 ...[/tex]

[tex]primes_{4} = ...[/tex]

Do you think these sequences behave similarly after a while? If they're all as "unpredictable", then that means that there are at least an infinity of sequences that are as unpredictable as the primes.

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# Other primes

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