A374449 Triangle read by rows: T(m,k) is the first number that starts a sequence of exactly k consecutive numbers with m prime factors, counted with multiplicity, if such a sequence is possible.

5, 2, 4, 9, 33, 8, 27, 170, 1083, 602, 2522, 211673, 16, 135, 1274, 4023, 12122, 204323, 355923, 6612470, 3405122, 49799889, 202536181, 3195380868, 5208143601

Offset: 1

Author

Robert Israel, Jul 08 2024

Keywords

• nonn
• tabf
• more

Comments

For m > 1, row m can have at most 2^m - 1 terms, because one out of every 2^m consecutive numbers is divisible by 2^m.

T(4,15) = A117969(4) = 97524222465.

Examples

			Triangle starts
  5 2
  4 9 33
  8 27 170 1083 603 3533 211673
T(3,2) = 27 because 27 = 3^3 and 28 = 2^2 * 7 each have 3 prime factors (counted with multiplicity) while 26 = 2*13 and 29 (prime) do not.
		

Crossrefs

Cf. A000079 (first column except for row 1), A115186, A113752, A117969 (last term in each row).

Programs

Maple

f:= proc(n)
uses priqueue;
local V,L, count, T, v, j, q, p, TP;
V:= Vector(2^n-1); count:= 0;
L:= [(-1)$(2^n),2^n];
initialize(pq);
insert([-2^(n),2$n],pq);
while count < 2^n-1 do
  T:= extract(pq); v:= -T[1];
  if L[-1] <> v-1 then
    for j from 1 while L[-1]-L[-j] = j-1 do
       if L[-j]-L[-j-1] <> 1 and V[j] = 0 then
         V[j]:= L[-j]; count:= count+1;
  fi od fi;
  L:= [op(L[2..-1]),v];
  q:= T[-1];
  p:= nextprime(q);
  for j from n+1 to 2 by -1 do
    if T[j] <> q then break fi;
    TP:= [T[1]*(p/q)^(n+2-j), op(T[2..j-1]), p$(n+2-j)];
    insert(TP,pq);
od od;
op(convert(V,list));
end proc:
f(1):= 5,2:
seq(f(i),i=1..3);

Formula

T(m,1) = 2^m for m >= 2.