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A219953 a(1) = 1; for n > 1, a(n) = smallest integer > a(n-1) such that a(n)*a(i)+1 is semiprime for all 1 <= i <= n-1.

%I A219953 #40 Feb 01 2023 01:22:04
%S A219953 1,3,8,38,86,318,504,3600,8132,83160,116850,202272,399126,6190086,
%T A219953 8756916,25253676,309709400,1112878446,1478724036,11062089360,
%U A219953 97331025386
%N A219953 a(1) = 1; for n > 1, a(n) = smallest integer > a(n-1) such that a(n)*a(i)+1 is semiprime for all 1 <= i <= n-1.
%C A219953 This is to A034881 as semiprimes A001358 are to primes A000040.
%C A219953 a(20) > 6*10^9. - _Giovanni Resta_, Jul 26 2015
%e A219953 a(1) = 1 by definition.
%e A219953 a(2) = 3: 3 > 1, and 1*3 + 1 = 4 = 2^2 is semiprime.
%e A219953 a(3) = 8: 8 > 3, and 1*8 + 1 = 9 = 3^2 is semiprime, and 3*8 + 1 = 25 = 5^2 is semiprime.
%e A219953 a(4) = 38: 38 > 8, and 1*38 + 1 = 39 = 3*13 is semiprime, and 3*38 + 1 = 115 = 5*23 is semiprime, and 8*38 + 1 = 305 = 5*61 is semiprime.
%e A219953 From _Michel Marcus_, Jul 26 2015: (Start)
%e A219953 The resulting semiprimes are:
%e A219953     4;
%e A219953     9,  25;
%e A219953    39, 115,  305;
%e A219953    87, 259,  689,  3269;
%e A219953   319, 955, 2545, 12085, 27349;
%e A219953   ...
%e A219953 (End)
%p A219953 A219953 := proc(n)
%p A219953     option remember;
%p A219953     if n= 1 then
%p A219953         1;
%p A219953     else
%p A219953         for a from procname(n-1)+1 do
%p A219953             issp := true ;
%p A219953             for i from 1 to n-1 do
%p A219953                 if numtheory[bigomega]( a*procname(n-i)+1) = 2 then
%p A219953                     ;
%p A219953                 else
%p A219953                     issp := false;
%p A219953                     break ;
%p A219953                 end if;
%p A219953             end do:
%p A219953             if issp then
%p A219953                 return a;
%p A219953             end if;
%p A219953         end do:
%p A219953     end if;
%p A219953 end proc: # _R. J. Mathar_, Dec 15 2012
%t A219953 a = {1}; Do[k = a[[n - 1]] + 1; While[! AllTrue[(k a[[n - #]] + 1) & /@ Range@ (n - 1), Total[Last /@ FactorInteger@ #] == 2 &], k++]; AppendTo[a, k], {n, 2, 13}]; a (* _Michael De Vlieger_, Jul 26 2015, Version 10 *)
%o A219953 (PARI) ok(v, n, k) = {v[n] = k; for (j=1, n-1, if (bigomega(1+v[n]*v[j]) != 2, return (0));); return (1);}
%o A219953 lista(nn) = {print1(k=1, ", "); v = [k]; for (n=2, nn, k = v[n-1]+1; v = concat(v, k); while (! ok(v, n, k), k++); v[n] = k; print1(k, ", "););} \\ _Michel Marcus_, Jul 26 2015
%Y A219953 Cf. A030063, A001358, A034881, A219761.
%K A219953 nonn,more
%O A219953 1,2
%A A219953 _Jonathan Vos Post_, Dec 01 2012
%E A219953 a(14)-a(17) from _Luke March_, Jul 26 2015
%E A219953 a(18)-a(19) from _Giovanni Resta_, Jul 26 2015
%E A219953 a(20)-a(21) from _Tyler Busby_, Jan 31 2023