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Lengths >= 2 of symmetrical subsequences within the prime gaps sequence.
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%I #60 Jan 24 2024 23:49:21

%S 2,3,5,5,3,9,5,2,3,3,3,5,3,3,5,2,11,2,3,3,2,3,2,3,2,3,5,3,2,3,3,3,3,3,

%T 3,3,3,3,2,3,3,2,5,2,2,3,7,3,2,3,3,5,5,7,3,3,5,2,2,3,5,3,3,3,2,5,2,3,

%U 2,2,3,7,3,3,2,3,3,3,2,3,3,3,3,3,3,2,3,2,3,3,3,3,2,3,3,3,3,3,3,3,2,3,3,5

%N Lengths >= 2 of symmetrical subsequences within the prime gaps sequence.

%C Points in the primes gap sequence (A001223) are taken successively at a term and halfway between terms.

%C The lengths here are of subsequences made of 2 or more symmetrically placed, consecutive prime gaps around such a point.

%C Some points only have a subsequence of length 0 or 1 around them and they are ignored.

%C Will all odd numbers appear in this sequence?

%C Do the terms have a long-term average?

%e The first lengths are as follows, around midpoints marked with ".",

%e Gaps: 1 2 2 4 2 4 2 = A001223

%e \_._/ length 2 = a(1)

%e \___.___/ length 3 = a(2)

%e \_______._______/ length 5 = a(3)

%o (PARI) diff(v) = vector(#v-1, i, v[i+1]-v[i]);

%o issym(v) = if (#v>1, for (j=1, #v\2, if (v[j] != v[#v-j+1], return(0))); return(1));

%o lista(nn) = my(v = diff(primes(nn))); for (len=2, #v, for (i=0, len\2, my(w = vector(len-2*i, j, v[i+j])); if (issym(w), print1(#w, ", "); break););); \\ _Michel Marcus_, Dec 05 2023

%Y Cf. A000040, A001223, A081235, A346399, A359440.

%K nonn

%O 1,1

%A _Tamas Sandor Nagy_, Dec 02 2023