[seqfan] sequences related to Sieve of Eratosthenes

Jamie Morken jmorken at shaw.ca
Sun Jan 14 08:57:24 CET 2018

```Hi,Considering the question: how many m > n aren't prime because of n?For example for n=2, knowing that all multiples of 2 aren't prime, the answer could be that 1/2 of all numbers m > n aren't prime because of 2.For n=3, knowing that all multiples of 3 aren't prime, and subtracting out
the multiples of 3 which are also multiples of 2, the answer could be that1/6 of all numbers m > n aren't prime because of 3.For non-prime n, no number m > n aren't prime because of n.For n=5, 1/15 of all numbers aren't prime because of 5.This is because 1/5 of m (the multiples of 5) aren't prime, andalso because 1/3 of those aren't prime as well due to 2 and 3 multiples,giving 1/3*1/5For n=7, 4/105 (simplified from 8/210) of all numbers aren't prime because of 7.This is because 1/7 of m (multiples of 7) aren't prime, and also because8/30 of those numbers aren't prime because of multiples of 2,3,5.For n=11, I haven't calculated this yet, however it may be x/2310, or x/210 * 1/11.These fractions approach zero for larger primes.For each prime n, the numbers that are not prime due to n, correspond to row's in http://oeis.org/A083140"Sieve of Eratosthenes arranged as an array and read by antidiagonals in
the up direction; n-th row has property that smallest prime factor is
prime(n)"
Some new possible sequences I found from considering how many m > n aren't prime because of n?
15,45,75,105,135,165 is for prime 5

35,175,245,385,455 is for prime 7

77,539,847,1001,1309 is for prime 11
143,… is for prime 13

221,… is for prime 17

The first column of the above sequences is in oeis: 15,35,77,143,221 (A006094 Products of 2 successive primes)The only sequence above I found when searching OEIS is 15,45,75,105,135,165:https://oeis.org/A143753Irregular triangle: A120070 read downwards antidiagonals.A120070 Triangle of numbers used to compute the frequencies of the spectral lines of the hydrogen atom.This is an interesting coincidence! :)Here is a picture from my spreadsheet for how I got those sequences:https://imgur.com/a/wESBJI think the other sequences are related so might be interesting but didn't find them in oeis.cheers,Jamie
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