%I #24 Aug 03 2018 15:44:53
%S 2,2,5,11,7,13,23,5,29,73,353,859,8423,92761,856717,126948763,
%T 122613509,2102184467,2530079,2659346387041447,334098941853251,
%U 148080802321968921649521449033,6863460080030077,1678153631189331624730247,7109839787546601453693131222417760271,5808110138398623777046165714073
%N a(1)=a(2)=2; thereafter, a(n)=gpf(1+a(n-1)a(n-2)), where gpf is greatest prime factor.
%t a = {2, 2}; Do[AppendTo[a, FactorInteger[1 + a[[n - 1]] a[[n - 2]]][[-1, 1]]], {n, 3, 26}]; a (* _Michael De Vlieger_, Aug 05 2015 *)
%t nxt[{a_, b_}] := {b, FactorInteger[a*b + 1][[-1, 1]]}; NestList[nxt,{2,2},30][[All,1]] (* _Harvey P. Dale_, Aug 03 2018 *)
%o (Sage)
%o def gpf(n):
%o return (factor(n)[-1])[0]
%o def A259828vec(m): # m>2=f
%o f=2
%o v=[2,2]
%o for i in range(f,m):
%o v.append(gpf(1+v[i-1]*v[i-2]))
%o return v
%o (PARI) gpf(n)=my(f=factor(n)[, 1]);f[#f];
%o first(m)=my(v=vector(m));v[1]=2;v[2]=2;for(i=3,m,v[i]=gpf(1+v[i-1]*v[i-2]));v;
%Y Cf. A260725, A000945.
%K nonn
%O 1,1
%A _Anders Hellström_, Aug 05 2015
%E Terms a(20)-a(26) from _Michael De Vlieger_, Aug 05 2015