[seqfan] Re: New sequence

Vladimir Shevelev shevelev at bgu.ac.il
Thu Apr 13 13:49:46 CEST 2017

Dear Seq Fans,

I wanted also for A284919 to find 
a simple additional condition such
that the corresponding subsequence
would grow very very fast or to 
be finite. It is not suitable a condition
of, say, n+2p-1 is composite or n+2p+1
is composite (recall that n is even
and p is prime such that n-p is also 
prime, while n+p is composite). 
Indeed, if n-3 is not prime and
n==2(mod 6), then p==n-p==1
(mod 3) (if, p==2 then n-p==0
(mod 3) which is contradiction).
But then n+2p-1==2+2-1==0
(mod 3) and in this case the 
condition that n+2p-1 is composite
is superfluos. In the close way
one can show that the condition
n+2p+1 is composite for many n 
is superfluos. So each such
"additional condition" gives only a
partial density reduction of A284919,
but not because the number of
representations of even n by sum
of two prime is less than one of 
odd n by prime and double prime.
Therefore, I considered an essential
condition for every even n>=4.
I choose  n+odd part of (p-1) is 
composite. In this case instead of
A284919 we obtain 0,2,118 and 
Peter Moses found only one more
term 868 up to 300000. I submitted
a comment in A284919.

Best regards,
From: SeqFan [seqfan-bounces at list.seqfan.eu] on behalf of Vladimir Shevelev [shevelev at exchange.bgu.ac.il]
Sent: 11 April 2017 13:37
To: Sequence Fanatics Discussion list
Subject: [seqfan] Re: New sequence

Thank you, dear Max! It is only
natural explanation.

Besides, I like  your negative
definition of A284919.
I submitted A277688 as a
simpler analog of A284919
for odd numbers which is
also connected with the
Lemoine-Levy conjecture.
It needs a continuation.

Best regards,

From: SeqFan [seqfan-bounces at list.seqfan.eu] on behalf of M. F. Hasler [seqfan at hasler.fr]
Sent: 10 April 2017 23:39
To: Sequence Fanatics Discussion list
Subject: [seqfan] Re: New sequence

On Mon, Apr 10, 2017 at 4:27 AM, Vladimir Shevelev <shevelev at bgu.ac.il>

> can be upgraded to "If there is
> another it is > 500001"

Dear Vladimir,
looking at http://oeis.org/A046927/graph suggests
that the number of decompositions 2n+1 = p +2q
grows strongly enough that it is increasingly improbable
that all of the resulting possibilities be composite.

PS1: the following PARI code checks this up to n=2*10^5 in about 2 seconds
and up to 5e5 in 8 seconds:


PS2: it would be interesting to add a comment which explains the "bands"
seen in http://oeis.org/A046927/graph.


I wanted to consider the sequence of odd n
> for which n+2p and n+2q are both composites
> for all pairs (p,q) such that p+2q=n.
> For example, 9 is not in the sequence,
> since 9+2*2=13, although 5+2*2=9.
> By handy I found that the first member
> is 59 (here the pairs (13,23), (37,11), (53,3))
> and all n+2p, n+2q are composites.
> Comparing to A284919, I was very surprised
> that up to 2*10^4 Peter Moses has found only
> more one term 151.

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