[seqfan] Re: Questions on A230492

[Vladimir Shevelev]

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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