# [seqfan] Re: Prime hatred

Eric Angelini Eric.Angelini at kntv.be
Thu Jan 7 17:59:47 CET 2010

```Many thanks, Franklin -- this is very clear.
Best,
É.

-----Message d'origine-----
De : seqfan-bounces at list.seqfan.eu
[mailto:seqfan-bounces at list.seqfan.eu]
De la part de franktaw at netscape.net
Envoyé : jeudi 7 janvier 2010 17:03
À : seqfan at list.seqfan.eu
Objet : [seqfan] Re: Prime hatred

I confirm these numbers.

I think it is nearly certain that the sequence contains no odd numbers
(except 1); you would have to get a large and increasing number of odd
non-primes from the (reverse) sums.  If there is another odd number in
the sequence, it is likely to be followed by a third odd number, then a
return to even numbers.  So I think it very, very unlikely that the
sequence contains all composite numbers.

It might well be that, if you change the initial 1 to a 2 (leaving the
rest of the sequence -- the values, not the definition -- unchanged),
you would get a permutation of the even numbers.  In fact, if there are
no more odd members, that seems nearly certain.

If it is the case tjat there are no other odd numbers in the sequence,
it is equivalent to "a(1) = 1; a(n) is the smallest even number not yet
used such that the cumulative sums are all non-prime".  Take the
constraint off that a(n) be even, and you get 1,3,2,4,5,6,7,8,9,...,
with the partial sums after the second being triangular numbers.  Just
requiring a(n) and the cumulative sums to be non-prime gives us a
sequence starting:

1, 8, 6, 9, 4, 10, 12, 14, 16, 15, 20, 18, 21, 22, 24, 25, 27, 26, 28,
30, 32, 34, 33, 35, 36, 38, 39, 40, 42, 45, 44, 46, 48, 49, 51, 50, 52,
54, 55, 56, 57, 58, 60, 62, 63, 65, 64, 66, 68, 70, 69, 72, 75, 74, 76,
77, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98,
99, 100, 102, 104, 105, 106, 108, 110, 111, 112, 114, 116, 115, 117,
118, 119, 120, 121, 122, 123, 124, 125, 126, 128, 129, 130, 132

This is also not in the OEIS.  It very probably is a permutation of the
non-primes.

-----Original Message-----
From: Robert Munafo <mrob27 at gmail.com>

Okay, after the joke reply and the 37 counterexample, I wrote a program
to
calculate the sequence and I get:

1, 8, 6, 10, 14, 12, 4, 20, 16, 24, 18, 22, 28, 26, 34, 30, 32, 36, 40,
42,
46, 38, 44, 52, 48, 54, 50, 58, 56, 62, 64, 60, 66, 68, 72, 70, 74, 80,
76,
78, 86, 82, 84, 90, 92, 94, 88, 98, 96, 104, ...

I am using the definition "The lexicographically first sequence of
natural
numbers such that no set of consecutive sequence members adds to a
prime,
and no number occurs in the sequence more than once."

...

On Thu, Jan 7, 2010 at 09:49, Eric Angelini <Eric.Angelini at kntv.be>
wrote:

>
> Hello SeqFans,
>
> « Build the lexicographically first sequence where no set
>  of consecutive integers sums up to a prime »
>
> I get:
>
> S = 1,8,6,10,12,18,4,14,22,20,16,24,26,...
>
> No prime in S, of course (a one-element set is possible)
>
> Could it be possible that S is a permutation of A018252?
> (the non-prime numbers)
>
> At this point of S there is no odd integer (except "1")...

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