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

[Eric Angelini]

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.
 
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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 ...
 
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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 ...
 
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Are those 4*2=8 possible seq worth the OEIS?
 
Best,
E.