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A357713 a(0) = 2; afterwards a(n) is the least prime greater than a(n-1) such that Omega(a(n-1) + a(n)) = n.

%I A357713 #28 Sep 25 2023 08:15:24
%S A357713 2,3,7,11,13,19,197,251,389,1531,2053,3067,17669,25339,66821,105211,
%T A357713 140549,318203,1008901,1940219,6710533,9804539,12215557,34970363,
%U A357713 49964293,75864827,276456709,864393979,1350198533,2877659899,4101661957,7709498107,16449692933,51196041979
%N A357713 a(0) = 2; afterwards a(n) is the least prime greater than a(n-1) such that Omega(a(n-1) + a(n)) = n.
%H A357713 Robert Israel, <a href="/A357713/b357713.txt">Table of n, a(n) for n = 0..2500</a>
%e A357713 2 + 3 = 5 (prime), 3 + 7 = 10 (semiprime), 7 + 11 = 18 (triprime).
%p A357713 f:= proc(n, a) # first prime b>a such that a+b is an n-almost-prime
%p A357713   uses priqueue;
%p A357713   local Aprimes, v, M, q, w;
%p A357713   M:= 10^100;
%p A357713   initialize(Aprimes);
%p A357713   insert([-2^n, 0, 2], Aprimes);
%p A357713   do
%p A357713     v:= extract(Aprimes);
%p A357713     if v[2] = n then
%p A357713       if -v[1] > 2*a and isprime(-v[1]-a) then return -v[1]-a fi
%p A357713     else
%p A357713       insert(v+[0, 1, 0], Aprimes);
%p A357713       q:= nextprime(v[3]);
%p A357713       w:= v[1]*(q/v[3])^(n-v[2]);
%p A357713       if w >= -M then insert([w, v[2], q], Aprimes) fi
%p A357713     fi
%p A357713   od
%p A357713 end proc:
%p A357713 R:= 2: a:= 2:
%p A357713 for n from 1 to 30 do
%p A357713   a:= f(n,a);
%p A357713   R:= R,a;
%p A357713 od:
%p A357713 R; # _Robert Israel_, Sep 19 2023
%t A357713 s = {p = 2}; Do[q = NextPrime[p]; While[k != PrimeOmega[p + q], q = NextPrime[q]]; AppendTo[s, p = q], {k, 30}]; s
%Y A357713 Cf. A001222 (Omega).
%K A357713 nonn
%O A357713 0,1
%A A357713 _Zak Seidov_, Oct 10 2022