[seqfan] Re: Questions on A230492

[David Wilson]

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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