[seqfan] Re: Add n to the composite a(n): the result is a composite too

[franktaw at netscape.net]

(1) Why do you make a special rule for n = 1?  The rest of the 
definition implies that a(1) = 8.  (And you can state that definition 
in one line, not three -- a(n) is the smallest composite number such 
that a(n) + n is also composite.)

(2) Shouldn't the sequence start with a(0) = 4?  (Again, no special 
rule required.)

(3) Compare A122984 and A122985.  If you submit these sequences, you 
should cross-ref to them.

(4) For the prime case, this is A020483 and A020484.

(5) I do think the first two sequences are worth submitting.  The cases 
where you constrain the sequences to be strictly increasing are much 
less interesting.  (Why do you call this "3bis"?  Or was this some sort 
of special symbol that got mangled by the software?  You should only 
use the basic ASCII/ANSI symbols in this mailing list and in the OEIS.)

Franklin T. Adams-Watters

-----Original Message-----
From: Eric Angelini <Eric.Angelini at kntv.be>

Hello SeqFans,

1-> a(1) = 8
2-> a(n) is composite
3-> a(n)+n is composite
4-> a(n+1) is the smallest possible integer

... this set of rules gives S1 (I think):

    S1 = 8 4 6 4 4  4  8  4  6  4  4  4  8  4  6  4  4  4  6  4 ...
     n = 1 2 3 4 5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 ...
a(n)+n = 9 6 9 8 9 10 15 12 15 14 15 16 21 18 21 20 21 22 25 24 ...

We could cut the three seq above in vertical slices, each one
showing the first pair of composites having n as difference.

---

If we insert the rule 3bis: a(n)<a(n+1), we get (I think):

    S2 = 8 10 12 14 15 16 18 20 21 22 24 26 27 28 30 32 33 34 35 ...
     n = 1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 ...
a(n)+n = 9 12 15 18 20 22 25 28 30 32 35 38 40 42 45 48 50 52 54 ...

S2 is not A080257; after 34, 35... S2 has 36, 39, 40, 42, ...
                              A080257 has 36, 38, 39, 40, 42, ...
... the term 38 is forbidden in S2 because 38 + n = 59, which is prime.

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Why not play accordingly with primes? We have to write new rules:

1-> a(1) = 3
2-> a(n) is prime
3-> a(n)+2n is prime
4-> a(n+1) is the smallest possible integer

... this new set of rules gives S3 (I think):

     S3 = 3 3  5  3  3  5  3  3  5  3  7  5  3  3  7  5  3  5  3  3 ...
     2n = 2 4  6  8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 ...
a(n)+2n = 5 7 11 11 13 17 17 19 23 23 29 29 29 31 37 37 37 41 41 43 ...

---

Again, if we add the 3bis rule a(n)<a(n+1), we get (I guess):

     S4 = 3  7 11 23 31 41 47 67 71  89 ...
     2n = 2  4  6  8 10 12 14 16 18  20 ...
a(n)+2n = 5 11 17 31 41 53 61 83 89 103 ...

---

Are those 4*2=8 possible seq worth the OEIS?