A120628 and record numbers of a087713
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Davide Rotondo
Dec 25, 2024, 11:09:33 AM12/25/24
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A120628
Primes such that their double is 1 away from a prime number.
2, 3, 5, 7, 11, 19, 23, 29, 31, 37, 41, 53, 79, 83, 89, 97, 113, 131, 139, 157, 173, 179, 191, 199, 211, 229, 233, 239, 251, 271, 281, 293, 307, 331, 337, 359, 367, 379, 419, 431, 439, 443, 491, 499, 509, 547, 577, 593, 601, 607, 619, 641, 653, 659, 661, 683 and
A087713
Greatest prime factor of the product of the neighbors of the n-th prime.
3, 2, 3, 3, 5, 7, 3, 5, 11, 7, 5, 19, 7, 11, 23, 13, 29, 31, 17, 7, 37, 13, 41, 11, 7, 17, 17, 53, 11, 19, 7, 13, 23, 23, 37, 19, 79, 41, 83, 43, 89, 13, 19, 97, 11, 11, 53, 37, 113, 23, 29, 17, 11, 7, 43, 131, 67, 17, 139, 47, 71, 73, 17, 31, 157, 79, 83, 13, 173, 29, 59,... if we take into consideration the sequence of record numbers of A087713 we get 3,5,7,11,19,23,29,31,37 ,41,53,79,83,89,97,... which is the sequence of numbers in A120628.
Primes such that their double is 1 away from a prime number.
2, 3, 5, 7, 11, 19, 23, 29, 31, 37, 41, 53, 79, 83, 89, 97, 113, 131, 139, 157, 173, 179, 191, 199, 211, 229, 233, 239, 251, 271, 281, 293, 307, 331, 337, 359, 367, 379, 419, 431, 439, 443, 491, 499, 509, 547, 577, 593, 601, 607, 619, 641, 653, 659, 661, 683 and
A087713
Greatest prime factor of the product of the neighbors of the n-th prime.
3, 2, 3, 3, 5, 7, 3, 5, 11, 7, 5, 19, 7, 11, 23, 13, 29, 31, 17, 7, 37, 13, 41, 11, 7, 17, 17, 53, 11, 19, 7, 13, 23, 23, 37, 19, 79, 41, 83, 43, 89, 13, 19, 97, 11, 11, 53, 37, 113, 23, 29, 17, 11, 7, 43, 131, 67, 17, 139, 47, 71, 73, 17, 31, 157, 79, 83, 13, 173, 29, 59,... if we take into consideration the sequence of record numbers of A087713 we get 3,5,7,11,19,23,29,31,37 ,41,53,79,83,89,97,... which is the sequence of numbers in A120628.
What do you think?
All the best
Davide
Robert Israel
Dec 25, 2024, 1:51:16 PM12/25/24
to seq...@googlegroups.com
It seems to be true.
If p is a prime > 3 such that 2*p-1 or 2*p+1 is a prime q, then p is the greatest prime factor of the neighbours of q, and is greater than any prime dividing a neighbour of any prime < q, so it is a record value in A087713.
On the other hand, we can't prove the converse, because we don't even have a proof that A120628 is infinite. We do know that A087713 is unbounded (for any prime p, Dirichlet's theorem shows there are infinitely many primes q == 1 mod p, so the greatest prime factor of q is at least p), so there are infinitely many record values there.
I checked the first 20000 record values in A087713, and all were in A120628.
Cheers,
Robert
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Davide Rotondo
Dec 25, 2024, 1:57:39 PM12/25/24
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Thank you very much Robert!
All the best
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