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

1,1

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.

Links

Table of n, a(n) for n=1..25.

Formula

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

Example

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.

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);

Crossrefs

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

Sequence in context: A124907 A181050 A020800 * A248261 A264991 A088507

Adjacent sequences: A374446 A374447 A374448 * A374450 A374451 A374452

Keyword

nonn,tabf,more

Author

Robert Israel, Jul 08 2024

Status

approved