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A116656 Slowest growing sequence of semiprimes having the semiprime-pairwise-sum property: for any i a(i)+a(j) is semiprime.

%I A116656 #20 May 29 2025 14:40:22
%S A116656 4,6,51,115,511,5263,116623,204091,823363,1144363,78325123,883337023,
%T A116656 6860264683,19613836423,167589841663
%N A116656 Slowest growing sequence of semiprimes having the semiprime-pairwise-sum property: for any i<j, a(i)+a(j) is semiprime.
%e A116656 Triangle of resulting semiprimes begins:
%e A116656       10
%e A116656       55,     57
%e A116656      119,    121,    166
%e A116656      515,    517,    562,    626
%e A116656     5267,   5269,   5314,   5378,   5774
%e A116656   116627, 116629, 116674, 116738, 117134, 121886
%t A116656 spQ[n_] := Plus @@ Last /@ FactorInteger[n] == 2; L = {0, 4}; Do[n = L[[-1]] + 1; While[! AllTrue[n + L, spQ], n++]; AppendTo[L, n], {9}]; Rest@ L (* _Giovanni Resta_, Jun 13 2018 *)
%o A116656 (PARI) lista(nn) = my(m, r, s, t, u, v=vector(nn=max(2, nn))); print1(v[1]=4, ", ", v[2]=6); for(n=3, nn, m=4; r=List([3]); forprime(p=2, oo, if(m*p>v[n-1], break); u=List([]); forprime(q=2, p-1, s=Set(v%(t=p*q)); for(i=1, #s, listput(u, Mod(t-s[i], t)))); s=List([]); for(i=1, #r, forstep(k=r[i], m*p, m, t=1; for(j=1, #u, if(k==u[j], t=0; break)); if(t, listput(s, k)))); r=s; m*=p); listsort(r); forstep(i=0, oo, m, for(j=1, #r, t=i+r[j]; if(t>v[n-1]&&bigomega(t)==2&&bigomega(t+4)==2&&bigomega(t+6)==2, for(k=3, n-1, if(!isprime((t+v[k])\2), t=0; break)); if(t, print1(", ", v[n]=t); break(2)))))); \\ _Jinyuan Wang_, May 29 2025
%Y A116656 Cf. A001358, A164979, A181620.
%K A116656 nonn,hard,more
%O A116656 1,1
%A A116656 _Zak Seidov_, Feb 21 2006
%E A116656 a(8)-a(10) from _R. J. Mathar_, Jan 23 2008
%E A116656 a(11)-a(12) from _Donovan Johnson_, Nov 11 2008
%E A116656 a(13) from _Donovan Johnson_, Jul 22 2011
%E A116656 a(14) from _Giovanni Resta_, Jun 13 2018
%E A116656 a(15) from _Giovanni Resta_, Jun 14 2018