# Composites, emirps, and emirpimes among de Polignac numbers (A006285)

Jonathan Post jvospost3 at gmail.com
Thu Jun 19 17:54:12 CEST 2008

```Sorry to email again so quickly to seqfans (there being reasonable
limits to number of emails per hour or day), but I neglected to
mention something else from my notes.

A050699  Numbers n such that n and n-reversed (<>n and no leading
zeros) have the same number of prime factors.

This should have added to its xrefs:
emirps (A006567)
emirpimes (A097393)
A109018  Least n-almost prime number which gives a different n-almost
prime number when digits are reversed.
A006567, A097393, A109023, A109023, A109024, A109025, A109026,
A109027, A109028, A109029, A109030, A109031.

My son, when he was 14, and in his 2nd year at university double
majoring in Math and Computer Science, independently discovered:
A050702  Numbers n such that n and n-reversed (<>n and no leading
zeros) have the same number of prime factors and these prime factors
(palindromes allowed here) are also reversals of each other.
We feel that  Patrick De Geest deserved the "nice" that this was granted.

Again, if one hates base sequences, one hates all of these. Nor would
many people want to see generalizations to bases other than 10.

But what is in OEIS is part of our domain of analysis, and we may use
our judgment to focus our efforts on core sequences on the one hand,
or out in the fringes or twilight zone.

On 6/19/08, Jonathan Post <jvospost3 at gmail.com> wrote:
> Roland Bacher and Andrew Weimholt make good points.
>
>  As to "'emirps' sequence hardly deserves mention in the first place"
>  -- is the issue the name (emirp = prime spelled backwards, because by
>  definition an Emirp is a prime whose reversal, base 10, is a different
>  prime)? Similarly, Emirpimes: numbers n such that n and its reversal
>  are distinct semiprimes.
>
>  That is, what repels people: the use of a base sequence, the name, or
>  the lack of conjectures?
>
>  As to conjectures, for A006285 Odd numbers not of form p + 2^x (de
>  Polignac numbers), I'd mentioned "Law of small numbers can fool one
>  into thinking them all noncomposite, as the first 13 are." My guess is
>  that the typical readers, spot checking the first few of the dozen
>  initial values of A006285 might fall into the trap of assuming them
>  all to be primes.  My surprise, after formulating and disproving that
>  conjecture, was to find out that many of the initial primes were a
>  special kind of primes, and the same with the composites.
>
>  Hence my wondering what the statistics are if one looks further.
>  There are b-lists for these sequences. What are the densities of
>  primes in de Polignac numbers?  Of emirps in primes?  And so forth.
>
>  The reason I'd emailed seqfans is that these preliminary results and
>  questions are not so interesting as to make it clear that there's
>  anything submittable, except perhaps a comment to A006285 that,
>  coincidentally, the first 13 values are noncomposite, but that
>  composites abound thereafter.
>
>  In deference to njas, I prefer NOT to submit a sequence unless I have
>  good reason to believe that someone will find it interesting or
>  useful.
>
>  That reticence is part of what I feel distinguishes me from some
>  others of "the usual suspects." Also, per my previous email which njas
>  said he liked, on guidelines for new submitters, that 148 of my seqs
>  so far hotlink to arXiv papers in Math, Physics, or Biology, with a
>  couple more in the pipeline.
>
>  To be a mathematician, with rare exception, means that one must READ
>  the Math papers out there, and have at least some ambition to write
>  and conventionally publish them. To read, as I've said, is to read
>  ACTIVELY, with pencil in hand, or calculator window open on the
>  browser, playing the the assumptions and boundaries of the sequence's
>  formulae and definition; and looking at other seqs cited (and the
>  papers or websites linked to).
>
>  The ways in which OEIS is a modern form of publication are very
>  interesting, and the co-evolution of OEIS and seqfans is a major test
>  of ideas about the future of Math and of publication.
>
>  Thank you very much for your active, honest, and kind responses,
>
>  -- Jonathan Vos Post
>
>
>  On 6/19/08, Andrew Weimholt <andrew at weimholt.com> wrote:
>  > On 6/18/08, Jonathan Post <jvospost3 at gmail.com> wrote:
>  >  >
>  >  > Again, I know that njas and many others dislike "base" sequences.
>  >  > But, to me, the intersection of two OEIS seqs is (if not too sparse) a
>  >  > legitimate submission.
>  >  >
>  >  > Anyone have an opinion on this? Or an extension?
>  >  >
>  >
>  >
>  > IMHO, for an intersection of two sequences to be worthy of being in
>  >  the OEIS requires...
>  >
>  >  (1) that the both sequences themselves be interesting and noteworthy
>  >  (2) that there be some evidence of a mathematical relationship between
>  >  the two sequences (other than that the intersection is not too
>  >  sparse).
>  >
>  >  Your proposed sequence meets neither of these, as the "emirps"
>  >  sequence hardly deserves mention in the first place, and so far there
>  >  is no grounds to make a meaningful conjecture about why the two
>  >  sequences should even be mentioned in the same breath.
>  >
>  >
>  >  Andrew
>  >
>

jvp> From seqfan-owner at ext.jussieu.fr  Thu Jun 19 17:33:33 2008
jvp> Date: Thu, 19 Jun 2008 08:32:07 -0700
jvp> From: "Jonathan Post" <jvospost3 at gmail.com>
jvp> To: "Andrew Weimholt" <andrew at weimholt.com>
jvp> Subject: Re: Composites, emirps, and emirpimes among de Polignac numbers (A006285)
jvp> Cc: "Sequence Fans" <seqfan at ext.jussieu.fr>
jvp> ...
jvp> As to conjectures, for A006285 Odd numbers not of form p + 2^x (de
jvp> Polignac numbers), I'd mentioned "Law of small numbers can fool one
jvp> into thinking them all noncomposite, as the first 13 are." My guess is
jvp> that the typical readers, spot checking the first few of the dozen
jvp> initial values of A006285 might fall into the trap of assuming them
jvp> all to be primes.  My surprise, after formulating and disproving that
jvp> conjecture, was to find out that many of the initial primes were a
jvp> special kind of primes, and the same with the composites.

If the cross-reference (Cf) in A006285 to the primes and composites (A098237)

%I A006285
%C A006285 Contains primes (A065381) and composites (A098237). - Jonathan Vos Post (jvospost3(AT)gmail.com), Jun 19 2008
%Y A006285 Cf. A133122, A098237, A065381.

Richard

jvp> From seqfan-owner at ext.jussieu.fr  Thu Jun 19 17:33:33 2008
jvp> Date: Thu, 19 Jun 2008 08:32:07 -0700
jvp> From: "Jonathan Post" <jvospost3 at gmail.com>
jvp> To: "Andrew Weimholt" <andrew at weimholt.com>
jvp> Subject: Re: Composites, emirps, and emirpimes among de Polignac numbers (A006285)
jvp> Cc: "Sequence Fans" <seqfan at ext.jussieu.fr>
jvp> ....
jvp> Hence my wondering what the statistics are if one looks further.
jvp> There are b-lists for these sequences. What are the densities of
jvp> primes in de Polignac numbers?  Of emirps in primes?  And so forth.

In the first, second, third etc group of 1000 Polignac numbers
(10 groups available in b006285.txt), the prime numbers are

0 0.408000 <= that is 40.8 percent primes in entries 1-1000 of A006285
1 0.367000 <= that is 36.7 percent primes in entries 1001-1999
2 0.339000
3 0.329000
4 0.322000
5 0.325000
6 0.309000
7 0.317000
8 0.289000
9 0.325000

The percentage of emirps (primes which remain prime after base-10 digit
reversal) is

0 0.088000  <= that is 8.8 percent emirps in the first 1000 entries of A006285
1 0.045000  <= that is 4.5 percent emirps in the 2nd 1000 entries
2 0.036000
3 0.107000
4 0.103000
5 0.100000
6 0.117000
7 0.013000
8 0.000000 <= no emirps in these 1000 entries
9 0.000000

In Maple:

BFILETOLIST := proc(bfilename)
end:

rev := proc(n)
end:

poli := BFILETOLIST("b006285.txt") ;

for t from 0 do
od:
for t from 0 do
od:

Jonathan Vos Post (I think) suggested recently that the OEIS should
have this sequence for all the major languages, and I fully agree.

I just got my colleague Vinay Vaishampayan to do the start
of the Hindi version (see A140395).  Here, as in many languages,
one needs a native speaker to say how to count the letters properly
(see the comments in this entry).

The OEIS already contains the following:

letters in n (in English): A005589*
letters in n (in other languages) (1): A001050 (Finnish), A001368 (Irish Gaelic), A003078 (Danish), A006968 or A092196 (Roman numerals), A007005 or A006969 (French), A006994 (Russian), A007208 (German), A007292 (Hungarian), A007485 or A090589 (Dutch),
letters in n (in other languages) (2): A008962 (Polish), A010038 (Czech), A011762 (Spanish), A027684 (Hebrew, dotted), A051785 (Catalan), A026858 (Italian), A056597 (Serbian or Croatian), A057435 (Turkish), A132984 (Latin), A140395 (Hindi),
letters in n (in other languages) (3): A053306 (Galego), A057696 (Brazilian Portuguese), A057853 (Esperanto), A059124 (Swedish), A030166, A112348, A112349 and A112350 (Chinese), A030166 (Japanese Kanji)

These are entries from the Index to the OEIS.  I may have missed some.