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March 18th, 2017, 05:17 PM   #1
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Similar sequence starts

How can I look up on OEIS, https://en.wikipedia.org/wiki/On-Lin...eger_Sequences, sequences that begin with similarly ordered matching integers?

Last edited by Loren; March 18th, 2017 at 05:19 PM.
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March 18th, 2017, 07:05 PM   #2
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For example:

1, 4, 7, 8, 10, 12, 15, 20...

1, 4, 7, 8, 10, 12, 13, 14...

for fairly long sequences of integers that are easily generated.

I believe there are formulas which "generate" primes for many integers before degenerating.
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March 18th, 2017, 08:21 PM   #3
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I think you just put in the common initial segment and it will show you as many distinct continuations as it knows about.

As far as formulas for finitely many primes, there are always the Lagrange polynomials. These let you fit n points to a degree n-1 polynomial. 3 point determine a quadratic and so forth. So f(1) = 2, f(2) = 3, f(3) = 5, etc. can always be fitted to a poly.

https://en.wikipedia.org/wiki/Lagrange_polynomial

And here's a really cool Lagrange polynomial calculator.

https://en.wikipedia.org/wiki/Lagrange_polynomial

For example I put in (1,2) (2,3) (3,5) (5,7) (6,11) (7,13) and it gave me back




Is that cool or what!

Of course we have to remember 6 rational coefficients to get back 6 primes. So having a polynomial that generates primes is less helpful than it might seem.
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Last edited by Maschke; March 18th, 2017 at 08:27 PM.
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March 18th, 2017, 09:21 PM   #4
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Which equation generates more primes?

Euler and Legendre are credited, respectively, with the following two polynomials which "generate" primes

N^2+N+41=p(N)

N^2-N+41=p(N)

where p is usually prime for a natural number N.

Would anyone here like to speculate on which equation generates more primes? (Watch out for those Big numbers!)
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