%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