login
Position of second appearance of 2n in the sequence of prime gaps A001223; if 2n does not appear at least twice, a(n) = -1.
2

%I #8 Aug 08 2022 15:54:44

%S 3,6,11,72,42,47,62,295,180,259,297,327,446,462,650,1315,1059,1532,

%T 4052,2344,3732,3861,8805,7234,4754,2810,4231,14124,5949,9834,17200,

%U 10229,19724,25248,15927,30765,42673,28593,24554,50523,44227,44390,29040,89715,47350

%N Position of second appearance of 2n in the sequence of prime gaps A001223; if 2n does not appear at least twice, a(n) = -1.

%C Prime gaps (A001223) are the differences between consecutive prime numbers. They begin: 1, 2, 2, 4, 2, 4, 2, 4, 6, ...

%t nn=1000;

%t gaps=Differences[Array[Prime,nn]];

%t mnrm[s_]:=If[Min@@s==1,mnrm[DeleteCases[s-1,0]]+1,0];

%t Table[Position[gaps,2*n][[2,1]],{n,mnrm[Select[Range[nn],Length[Position[gaps,2*#]]>=2&]]}]

%Y The position of the first (instead of second) appearance of 2n is A038664.

%Y Column k = 2 of A356222.

%Y The position of the n-th appearance of 2n is A356223.

%Y A001223 lists the prime gaps, reduced A028334.

%Y A073491 lists numbers with gapless prime indices.

%Y A274121 counts appearances of the n-th prime gap in those prior.

%Y A356226 gives the lengths of maximal gapless intervals of prime indices.

%Y Cf. A029709, A066205, A137921, A193829, A287170, A328335, A328457, A356224, A356225.

%K nonn

%O 1,1

%A _Gus Wiseman_, Aug 02 2022