# [seqfan] Re: Palindromic Subsequences Of Prime Differences

franktaw at netscape.net franktaw at netscape.net
Fri Mar 26 01:11:59 CET 2010

```a(4) = p(2136) = 18713; the primes are 18713 18719 18731 18743 18749,
with differences 6,12,12,6.

In general, the even values are going to be bigger, because all the
differences for even n must be multiples of 3 (and since almost all
prime differences are even, they must be multiples of 6). Consider the
middle two terms in the sequence of differences. If these are m, and
the prime right before them is p, we have a sequence of three primes p,
p+m, p+2m. If m is not divisible by 3, one of these 3 "primes" must
itself be divisible by 3 -- which is possible only for the n=2 case.
Now look at the next pair from the middle. Skipping a few steps, we get
a sequence of primes p, p+m, p+m+6j, p+m+12j, p+2m+12j; and again, m
must be divisible by 3 to avoid having at least one of these divisible
by 3. Continuing in this way, we conclude that all the differences must
be multiples of 3.

It is obvious that the bisections of this sequence are almost
monotonically increasing: since a sequence with n+2 differences has a
sequence of n differences in the middle, if a(n) = p(k), a(n+2) >=
p(k-1). Are these bisections actually strictly increasing? If we look
instead at the last prime in each finite sequence, 3,7,13,18749,43,...,
it is easy to see that the bisections of this sequence are strictly
increasing.

The sequence of prime indices, 1,2,3,2136,4,... is also not in the
OEIS; nor are either of the sequences based on the largest member of
each finite sequence instead of the smallest.

-----Original Message-----
From: Leroy Quet <q1qq2qqq3qqqq at yahoo.com>

I am wondering if this sequence is in the EIS already. It doesn't look
like it
(searching using wildcards).

a(n) = the smallest prime p(k) (the k-th prime) such that:
p(k+j) - p(k+j-1) = p(n+k+1-j) - p(n+k-j),
for all j where 1 <= j <= n.

I get the sequence beginning:
(offset 1)

2, 3, 5, _, 7, _, 17

As an example: List the 8 primes starting with 17:
17,19,23,29,31,37,41,43

List the 7 differences between these consecutive primes:
2,4,6,2,6,4,2

Since this is a palindromic finite sequence, and since the sequence of
8 primes
starting with 17 are the smallest-valued string of 8 primes having this
property, then a(7) = 17.
-

First of all, I can't find a value for a(4) by checking all 97
differences
between the primes in the "list" link of sequence A001223.
Is there even a prime where p(k+1) - p(k) = p(4+k)-p(3+k), and  p(k+2)
- p(k+1)
= p(3+k)-p(2+k)?

Sorry that I am so dense, but this must be obvious.

And if a(4) exists, does a(n) exist for all n's?

And, oh yeah, can someone please extend the sequence?

Thanks,
Leroy Quet

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