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.

3, 91, 66, 340, 2548, 30940, 67228, 6290620, 81818748, 1336727934, 19729482496, 358398854656, 1934923637500, 115877891562496

Offset: 1

Author

Zak Seidov and Robert Israel, Jul 07 2024

Keywords

• nonn
• hard
• more

Comments

For n >= 3, a(n) <= 2 * A113752(n-1), with equality when a(n) is even.

a(15) <= 2495158931562496. - Martin Ehrenstein, Jul 11 2024

Examples

			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.
		

Crossrefs

Cf. A001222, A113752.

Programs

Maple

f:= proc(m) uses priqueue;
local S, pq, T, v, TP, q, p, j;
  S:= {-10,-9,-8,-7};
initialize(pq);
insert([-2^m,2$m],pq);
do
  T:= extract(pq); v:= -T[1];
  if {v-2,v-4} subset S then return v-4 fi;
  S:= (S minus {min(S)}) union {v};
  q:= T[-1];
  p:= nextprime(q);
  for j from m+1 to 2 by -1 do
      if T[j] <> q then break fi;
    TP:= [T[1]*(p/q)^(m+2-j),op(T[2..j-1]),p$(m+2-j)];
      insert(TP, pq)
od od;
end proc:
map(f, [$1..11]);

Extensions

a(12)-a(14) from Martin Ehrenstein, Jul 11 2024