[seqfan] Re: A073674.
israel at math.ubc.ca
israel at math.ubc.ca
Tue Dec 18 09:11:16 CET 2012
For any even positive integers a_1, a_2, ..., a_n, there are infinitely
many even positive integers t such that a_1 a_2 ... a_n t + 1 is prime:
this follows from Dirichlet's theorem on primes in arithmetic progressions.
So you could define a_{n+1} as the least such t that is not in {a_1, a_2,
..., a_n}. As far as I know there is no guarantee that this leads to a
permutation, i.e. there might be some even integer that never appears in
the sequence. However, if the partial products a_1 ... a_n grow like 2^n
n!, heuristically the probability of a_1 ... a_n t + 1 being prime is on
the order of 1/log(a_1 ... a_n) ~ 1/(n log n), and since sum_n 1/(n log n)
diverges we might expect that there should be infinitely many n for which
this is true.
If you want to submit this as a sequence, I would suggest leaving
"permutation" out of the definition, but putting in a comment about the
issue.
By the way, the first 100 a_n are 2, 6, 8, 12, 16, 10, 4, 30, 26, 22, 24,
14, 50, 42, 18, 64, 46, 60, 32, 36, 20, 34, 28, 108, 48, 44, 68, 282, 90,
54, 76, 62, 180, 66, 132, 86, 74, 38, 58, 106, 120, 52, 244, 94, 100, 82,
138, 156, 98, 72, 172, 150, 248, 154, 166, 114, 162, 126, 124, 208, 222,
324, 212, 206, 178, 104, 238, 232, 78, 210, 170, 184, 92, 88, 392, 122,
286, 116, 194, 84, 446, 218, 144, 140, 130, 40, 326, 70, 254, 490, 276,
234, 380, 56, 484, 202, 174, 198, 136, 272
The first even integer that hasn't appeared yet is 80.
Robert Israel
University of British Columbia
On Dec 17 2012, Ed Jeffery wrote:
>Consider "the" lexicographically smallest permutation of the even natural
>numbers such that each partial product with 1 added to it is equal to a
>prime. Then the sequence could begin as
>
>2, 6, 8, 28, 4, 26, 10, ...,
>
>with the corresponding primes being
>
>3, 13, 97, 2689, 10753, 279553, 2795521....
>
>Neither sequence is in OEIS (nor is {1, 3, 4, 14, 2, 13, 5,...}).
>
>But, given the rate of growth (seems to be > 2^n*n!) of this sequence of
>primes (each k-th partial product is essentially a product of k distinct
>natural numbers multiplied by 2^k), how could such an assertion be proved,
>since we soon run out of computable or known primes where factorization is
>then, well, a factor?
>
>This leads me to the same question regarding A073674. I have searched for a
>proof of this sequence but cannot find one. So is there a proof somewhere?
>
>http://oeis.org/A073674
>
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