My Math Forum New sequence containing all the primes + some semi-prime

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February 23rd, 2010, 11:16 AM   #11
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Re: New sequence containing all the primes + some semi-prime

Quote:
 Originally Posted by momo I said and I quote myself "Are all the prime numbers >3 in the sequence and why? "
Yes, and I've twice explained why they are.

Quote:
 Originally Posted by momo Anyway there is other ways to sieve the sequence such as it removes all non primes.
Sure, Eratosthenes came up with a good method a while back.

Quote:
 Originally Posted by momo 35 is not divisible by 2 nor by 3 35 is a semi-prime 35 is an element of the sequence
Yes, I already told you that all semiprimes not divisible by 2 or 3 are in the sequence.

February 23rd, 2010, 11:19 AM   #12
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Re: New sequence containing all the primes + some semi-prime

Quote:
Originally Posted by CRGreathouse
Quote:
 Originally Posted by momo I said and I quote myself "Are all the prime numbers >3 in the sequence and why? "
Yes, and I've twice explained why they are.

Quote:
 Originally Posted by momo Anyway there is other ways to sieve the sequence such as it removes all non primes.
Sure, Eratosthenes came up with a good method a while back.

Quote:
 Originally Posted by momo 35 is not divisible by 2 nor by 3 35 is a semi-prime 35 is an element of the sequence
Yes, I already told you that all semiprimes not divisible by 2 or 3 are in the sequence.
14 is semi-prime divisible by 2
22 and so on
14, 22,38 etc.... are on the sequence.

February 23rd, 2010, 11:25 AM   #13
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Re: New sequence containing all the primes + some semi-prime

Quote:
 Originally Posted by momo 14 is semi-prime divisible by 2 22 and so on 14, 22,38 etc.... are on the sequence.
I never claimed that they weren't in the sequence.

February 23rd, 2010, 11:30 AM   #14
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Re: New sequence containing all the primes + some semi-prime

Quote:
Originally Posted by CRGreathouse
Quote:
 Originally Posted by momo 14 is semi-prime divisible by 2 22 and so on 14, 22,38 etc.... are on the sequence.
I never claimed that they weren't in the sequence.

The sequence contains:

All primes except those divisible by {2, 3}
All semiprimes except those divisible by {2, 3}
All 3-almost primes except those divisible by {2, 3}
...
All k-almost primes except those divisible by {2, 3}

So 14,22,38 are divisible by 2
14,22,38 are semi-prime
I do not understand ....

February 23rd, 2010, 12:20 PM   #15
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Re: New sequence containing all the primes + some semi-prime

Quote:
 Originally Posted by momo Here is your claim The sequence contains: All primes except those divisible by {2, 3} All semiprimes except those divisible by {2, 3} All 3-almost primes except those divisible by {2, 3} ... All k-almost primes except those divisible by {2, 3}
Yes, I stand by that claim.

Quote:
 Originally Posted by momo So 14,22,38 are divisible by 2 14,22,38 are semi-prime I do not understand ....
I didn't say that there are no semiprimes in the sequence that are divisible by 2 or 3. I said that all the ones that aren't are in the sequence.

"All primes are natural numbers" does not imply "No composites are natural numbers". "All numbers not divisible by 2 or 3 are in the sequence" does not imply "No numbers divisible by 2 or 3 are in the sequence".

 February 23rd, 2010, 04:00 PM #16 Senior Member   Joined: Nov 2007 Posts: 633 Thanks: 0 Re: New sequence containing all the primes + some semi-prime I'm really dumb!
February 24th, 2010, 05:11 AM   #17
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Re: New sequence containing all the primes + some semi-prime

Quote:
 Originally Posted by momo I'm really dumb!
Nah, but you missed something. Maybe you can now prove precisely which even semiprimes are in the sequence. (I did this, but you can do it on your own. You'll need to consider four cases: 4, 6, 8k+2 with k > 0, and 8k+6 with k > 1.) Here's the formula for S:
$S(n)=\frac{n(n-1)(n+1)}{12}$ for n odd and $S(n)=\frac{n(n+2)(2n-1)}{24}$ for n even.

Maybe you can even prove which semiprimes divisible by 3 are in the sequence. I haven't done that.

 Tags primes, semiprime, sequence

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