[seqfan] Re: New sequence

Vladimir Shevelev shevelev at bgu.ac.il
Sun Apr 9 21:43:22 CEST 2017

Dear Seq Fans,

I wanted to create an analog of the beautiful
sequence A284919 by Claudio Meller for odd
numbers. I remembered another known 
Levy conjecture that every odd number n>=7
is sum of a prime and double of a prime:
n=p+2q, where p>=3, q>=2.
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.

I ask anyone to obtain again this astonishing 

P.S. Maybe, for example, for analogous sequence
with the conditions for odd n: all n+p-1, n+2q
are composites as n=p+2q, we have more
terms up to 2*10^4? However, here by handy
I found that the first member is 83.

Best regards,
From: SeqFan [seqfan-bounces at list.seqfan.eu] on behalf of M. F. Hasler [seqfan at hasler.fr]
Sent: 06 April 2017 03:10
To: Sequence Fanatics Discussion list
Subject: [seqfan] Re: New sequence

On Wed, Apr 5, 2017 at 5:22 PM, Graeme McRae <graememcrae at gmail.com> wrote:

> for all possible primes p1, p2 such that p1+p2=E,
> neither E+p1 nor E+p2 is prime.
> The first few terms I found were:
> 4, 6, 28, 38, 52, 58, 62, 68, 74, 80, 82, 88, 94, 98, 112, 118, 122, 124,
> 128, 136...

I searched for it in OEIS, and did not find it.

I submitted https://oeis.org/draft/A284919
including PARI code based on Charles' proposal,
but not Graeme McRae's comment on the probably finite "for all" (or: no...)
(cf below: you are invited to add it, conveniently rephrased).
I also chose to include the two initial terms 0 and 2 which may be arguable
but may affect search results only in a positive sense.

Looking at other Goldbach related sequences, they often use n where E=2n,
so I wonder whether the sequence a(n)/2 should also be submitted.

> I also looked for even numbers such that *all* the values E+pn are prime.
> The only such even numbers I found smaller than 1000 were 8 and 12. In
> fact, the only even numbers smaller than 1000 such that all but one of the
> values E+pn are prime are 8, 12, 18, 24, and 30. I wonder if these
> sequences are finite.

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