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A335667 a(n) is the smallest even number k such that k-1 and k+1 are both n-almost primes.

%I A335667 #19 Jul 10 2023 07:33:45
%S A335667 4,34,274,2276,8126,184876,446876,18671876,95234374,1144976876,
%T A335667 6018359374,281025390626,2068291015624,6254345703124,181171630859374,
%U A335667 337813720703126,31079046044921876,205337027587890626
%N A335667 a(n) is the smallest even number k such that k-1 and k+1 are both n-almost primes.
%C A335667 10^13 < a(15) <= 181171630859374. - _Giovanni Resta_, Jun 21 2020
%H A335667 David A. Corneth, <a href="/A335667/a335667.gp.txt">Upper bounds on a(n) for n = 1..37 where values with * are proven values</a>
%e A335667 a(1) = 4 since 4 - 1 and 4 + 1 are both primes.
%e A335667 a(2) = 34 since 34 - 1 = 33 = 3*11 and 34 + 1 = 35 = 5*7 are both semiprimes.
%e A335667 a(3) = 274 since 274 - 1 = 273 = 3*7*13 and 274 + 1 = 275 = 5^2 * 11 are both 3-almost primes.
%t A335667 m = 8; v = Table[0, {m}]; c = 0; o1 = 1; n = 4; While[c < m, o2 = PrimeOmega[n + 1]; If[o1 == o2 && v[[o1]] == 0, c++; v[[o1]] = n]; o1 = o2; n += 2]; v
%o A335667 (PARI)
%o A335667 generate(A, B, n, k) = A=max(A, 2^n); (f(m, p, n) = my(list=List()); if(n==1, forprime(q=max(p, ceil(A/m)), B\m, if(bigomega(m*q+2) == k, listput(list, m*q+1))), forprime(q=p, sqrtnint(B\m, n), list=concat(list, f(m*q, q, n-1)))); list); vecsort(Vec(f(1, 3, n)));
%o A335667 a(n) = my(x=2^n, y=2*x); while(1, my(v=generate(x, y, n, n)); if(#v >= 1, return(v[1])); x=y+1; y=2*x); \\ _Daniel Suteu_, Jul 10 2023
%Y A335667 Cf. A001222, A088077, A115186, A154704.
%K A335667 nonn,more
%O A335667 1,1
%A A335667 _Zak Seidov_ and _Amiram Eldar_, Jun 17 2020
%E A335667 a(12)-a(14) from _Giovanni Resta_, Jun 21 2020
%E A335667 a(15)-a(18) from _Daniel Suteu_, Jul 10 2023