[seqfan] Re: Questions on A230492

[israel at math.ubc.ca]

I found no more pairs of primes for n <= 4440.
p(n) is the closest integer to 1/5*(sqrt(5)-1)*(7/2+3/2*sqrt(5))^n)-1
and q(n) is the closest integer to 2/5*(7/2+3/2*sqrt(5))^n.
Heuristically, the probability that p(n) and q(n) are both prime
is on the order of 1/n^2, so since sum_n 1/n^2 is finite we would
expect only finitely many such n.  

Robert Israel
University of British Columbia and D-Wave Systems


On Dec 26 2013, David Wilson wrote:

>I think it's likely that there are other squares in this sequence.
>
> 209^2 = 11^2*19^2 has prime signature p^2*q^2, so perhaps we should look 
> for other elements with this signature.
>
>p^2*q^2 will be in A230492 if it satisfies the critical equality
>
>[1]  1 + p + q + p^2 + p*q = q^2.
>
>Solving [1] empirically for integer (p, q), we get solutions
>
>(1, 3)
>(11, 19)
>(79, 129)
>(545, 883)
>(3739, 6051)
>(25631, 41473)
>(175681, 284259)
>(1204139, 1948339)
>(8253295, 13354113)
>(56568929, 91530451)
>...
>
>The theory of quadratic Diophantine equations tells us that these solutions
>will satisfy a set of linear recurrences. In this case we find that
>
>p(n) = 7p(n-1) - p(n-2) + 3.
>q(n) = 7q(n-1) - q(n-2) - 1.
>
>or, if you like pure recurrences
>
>p(n) = 8p(n-1) - 8p(n-2) + p(n-3).
>q(n) = 8q(n-1) - 8q(n-2) + q(n-3).
>
> With these recurrences, we can quickly generate (p, q) pairs satisfying 
> [1].
>
> With the tools at my disposal, I was only able to test q < 2^64, and in 
> that range (p, q) = (11, 19) was the only pair of primes I found. 
> However, the divisor patterns do not seem to rule out a larger pair of 
> primes (p, q) for which p^2*q^2 would be in A230492. Perhaps a seqfan 
> with power tools could find such a pair.
>
>> -----Original Message-----
>> From: SeqFan [mailto:seqfan-bounces at list.seqfan.eu] On Behalf Of Vladimir
>> Shevelev
>> Sent: Wednesday, December 25, 2013 1:11 PM
>> To: Sequence Fanatics Discussion list
>> Subject: [seqfan] Re: Questions on A230492
>> 
>> Dear Michel and SeqFans,
>> 
>> I would like to pay attention to one more question.
>> Using b-file of A230492 containing 250 terms, I found among them only one
>> square: a(19)=43681=209^2. Does exist the next one?
>> It is clear, that square of a prime is not in the sequence.
>> In case of a semiprime p*q, it is easy to see that p and q should be
>dstinct.
>> Let p<q. Then, it is easy to see that, a key equality is
>q^2-1=(p+1)*(p+q).
>> For example, for p=11, q=19, we have the equality 360=12*30.
>> Therefore, (11*19)^2 is in the sequence.
>> 
>> 
>> Regards,
>> Vladimir
>> 
>> ________________________________________
>> From: SeqFan [seqfan-bounces at list.seqfan.eu] on behalf of
>> michel.marcus at free.fr [michel.marcus at free.fr]
>> Sent: 19 December 2013 18:16
>> To: seqfan at list.seqfan.eu
>> Subject: [seqfan] Questions on A230492
>> 
>> Hi SeqFans,
>> 
>> I have recently extended A230492 to 250 terms. Unless mistaken, it still
>verify
>> the property that even terms are perfect.
>> I have also checked that the 5 first perfect numbers (A000396) belong to
>this
>> sequence (the 6th one is taking some time).
>> 
>> Is it possible to prove that perfect numbers are in A230492, but no other
>> even numbers ?
>> And what could be said about the odd terms ?
>> 
>> I have searched the OEIS and found only 2 other sequences with the same
>> property:
>> A034897   Hyperperfect numbers.
>> A225417   Composite numbers which contain their sum of aliquot parts as a
>> substring.
>> Are there others ?
>> 
>> Thank you for your help.
>> Michel
>> 
>> 
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