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%I #26 Oct 09 2021 07:40:40
%S 1,7,11,91,164,931,5131,6403,11811,19507,87907,87914,341811,7961923,
%T 7963880,8037362,8044275,8140244,8593387,11055787,14017267,37262755,
%U 164218555,551667835,2340626587,2970952034,3788070787
%N The lexicographically earliest sequence with a(1) = 1 and Omega(a(i) + a(k)) = 3 for i < k <= n.
%C Sum of any two terms is in A014612. Is the sequence infinite?
%e 1 + 7 = 8 = A014612(1), 1 + 11 = 12 = A014612(2), 7 + 11 = 18 = A014612(3).
%t a[1] = 1; a[n_] := a[n] = Module[{k = a[n - 1] + 1}, While[! AllTrue[Array[a, n - 1], PrimeOmega[# + k] == 3 &], k++]; k]; Array[a, 12] (* _Amiram Eldar_, Sep 29 2021 *)
%o (PARI) first(n)=my(v=vector(n),t); v[1]=1; for(k=2,n, t=v[k-1]; while(t++, for(i=1,k-1, if(bigomega(v[i]+t)!=3, next(2))); break); v[k]=t); v; \\ _Charles R Greathouse IV_, Oct 03 2021
%o (PARI) nxt(v,n=v[#v]+1)=my(b=min(97,n-1)); while(1, for(i=1,#v, my(f=factor(v[i]+n,b),e=f[,2]); if(vecsum(e)>3 || vecsum(e)==3 && !isprime(f[#e,1]), n++; next(2))); for(i=1,#v, if(bigomega(v[i]+n)!=3, n++; next(2))); return(n)); \\ _Charles R Greathouse IV_, Oct 03 2021
%Y Cf. A001222 (Omega), A014612.
%K nonn
%O 1,2
%A _Zak Seidov_, Sep 26 2021