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A181620 Sequence starting with 2 such that the sum of any two distinct terms is a semiprime having two distinct prime factors.

%I A181620 #21 Jun 02 2025 22:16:56
%S A181620 2,4,31,91,183,4411,29611,59935,110791,10418851,658653031,20123369491,
%T A181620 518294316451,947137685251
%N A181620 Sequence starting with 2 such that the sum of any two distinct terms is a semiprime having two distinct prime factors.
%C A181620 Choose the first number not leading to a contradiction.
%C A181620 The sequence starting with 1 is finite: {1, 5, 9, 86, 212};
%C A181620 Sequence starting with 3: {3, 7, 19, 32, 55, 246, 39499, ...};
%C A181620 Sequence starting with 4: {4, 6, 29, 89, 137, 749, 1685, 16497, ...}.
%e A181620 The subset {2, 4, 31} produces the three sums {6, 33, 35} which factor as {2*3, 3*11, 5*7}.
%p A181620 with(numtheory):nn:=500000:T:=array(1..nn): U:=array(1..nn): for p from 1 to
%p A181620   nn do: T[p]:=p+1:U[p]:=2:od:for u from 1 to 10 do: k:=1+u:for n from u+1 to
%p A181620   nn do:s:=T[n]+T[u]:s1:=nops(factorset(s)):s2:=bigomega(s):if s1=2 and s2=2 then
%p A181620   U[k]:=T[n]:k:=k+1:else fi:od:for i from 1 to nn do:T[i]:=U[i]:od:od:for j from
%p A181620   1 to 30 do:print( T[j]):od:
%t A181620 TwoDistinct[n_]:=Module[{p,e}, {p,e}=Transpose[FactorInteger[n]]; Length[p]==2 && e=={1,1}]; t={2}; k=2; Do[While[k++; !And@@TwoDistinct/@(k+t)]; AppendTo[t,k], {6}]; t
%Y A181620 Cf. A180514, A180565, A180615.
%K A181620 nonn,more
%O A181620 1,1
%A A181620 _Michel Lagneau_, Jan 31 2011
%E A181620 Removed 84835 and a(10)-a(12) from _Donovan Johnson_, Feb 14 2011
%E A181620 a(13)-a(14) from _Jinyuan Wang_, May 29 2025