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A374392 a(n) is the least number k such that k, k + 2 and k + 4 all have exactly n prime factors, counted with multiplicity.

%I A374392 #10 Jul 11 2024 16:26:52
%S A374392 3,91,66,340,2548,30940,67228,6290620,81818748,1336727934,19729482496,
%T A374392 358398854656,1934923637500,115877891562496
%N A374392 a(n) is the least number k such that k, k + 2 and k + 4 all have exactly n prime factors, counted with multiplicity.
%C A374392 For n >= 3, a(n) <= 2 * A113752(n-1), with equality when a(n) is even.
%C A374392 a(15) <= 2495158931562496. - _Martin Ehrenstein_, Jul 11 2024
%e A374392 a(3) = 66 because 66 = 2 * 3 * 11, 68 = 2^2 * 17 and 70 = 2 * 5 * 7 all have 3 prime factors, counted with multiplicity, and 66 is the least number that works.
%p A374392 f:= proc(m) uses priqueue;
%p A374392 local S, pq, T, v, TP, q, p, j;
%p A374392   S:= {-10,-9,-8,-7};
%p A374392 initialize(pq);
%p A374392 insert([-2^m,2$m],pq);
%p A374392 do
%p A374392   T:= extract(pq); v:= -T[1];
%p A374392   if {v-2,v-4} subset S then return v-4 fi;
%p A374392   S:= (S minus {min(S)}) union {v};
%p A374392   q:= T[-1];
%p A374392   p:= nextprime(q);
%p A374392   for j from m+1 to 2 by -1 do
%p A374392       if T[j] <> q then break fi;
%p A374392     TP:= [T[1]*(p/q)^(m+2-j),op(T[2..j-1]),p$(m+2-j)];
%p A374392       insert(TP, pq)
%p A374392 od od;
%p A374392 end proc:
%p A374392 map(f, [$1..11]);
%Y A374392 Cf. A001222, A113752.
%K A374392 nonn,hard,more
%O A374392 1,1
%A A374392 _Zak Seidov_ and _Robert Israel_, Jul 07 2024
%E A374392 a(12)-a(14) from _Martin Ehrenstein_, Jul 11 2024