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A255609 a(1)=2; a(n) = the smallest prime p such that a(n)-a(n-1) is semiprime (A001358).

%I A255609 #18 Dec 29 2022 09:49:05
%S A255609 2,11,17,23,29,43,47,53,59,73,79,83,89,103,107,113,127,131,137,151,
%T A255609 157,163,167,173,179,193,197,211,233,239,277,281,307,311,317,331,337,
%U A255609 347,353,359,373,379,383,389,463,467,541,547,557,563,569,607
%N A255609 a(1)=2; a(n) = the smallest prime p such that a(n)-a(n-1) is semiprime (A001358).
%C A255609 Sequence with any initial prime term a(1) eventually merges with this sequence: 3,7,11; 5,11; 13,17; 19,23; 31,37,41,47.
%C A255609 For n > 1, a(n) = A289750(n+1). - _Jon E. Schoenfield_, Nov 26 2017
%H A255609 Robert Israel, <a href="/A255609/b255609.txt">Table of n, a(n) for n = 1..10000</a>
%e A255609 a(2) - a(1) = 11 - 2 = 9 = 3*3;
%e A255609 a(3) - a(2) = 17 - 11 = 6 = 2*3;
%e A255609 a(81) - a(80) = 1009 - 887 = 122 = 2*61.
%p A255609 A:= Vector(100): A[1]:= 2:
%p A255609 for n from 2 to 100 do
%p A255609   p:= A[n-1];
%p A255609   do
%p A255609     p:= nextprime(p);
%p A255609     until numtheory:-bigomega(p-A[n-1]) = 2;
%p A255609   A[n]:= p;
%p A255609 od:
%p A255609 convert(A,list); # _Robert Israel_, Dec 28 2022
%t A255609 s = {2}; p = 2; Do[q = NextPrime[p]; While[2 != PrimeOmega[q - p], q = NextPrime[q]]; AppendTo[s, q]; p = q, {100}]; s
%t A255609 sp[n_]:=Module[{p=NextPrime[n]},While[PrimeOmega[p-n]!=2,p= NextPrime[ p]];p]; NestList[sp,2,60] (* _Harvey P. Dale_, Oct 10 2015 *)
%o A255609 (PARI) v=[2];forprime(p=3,300,if(bigomega(p-v[#v])==2,v=concat(v,p)));v \\ _Derek Orr_, Feb 28 2015
%Y A255609 Cf. A001358, A156829, A289750.
%K A255609 nonn
%O A255609 1,1
%A A255609 _Zak Seidov_, Feb 28 2015