A394211 Triangle read by rows: T(n, k) is the number of partitions of n into parts having exactly k prime factors (counted with multiplicity), with 0 <= k <= floor(log_2(n)) for n >= 1 and k = 0 for n = 0.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 0, 1, 2, 1, 1, 3, 0, 1, 3, 1, 1, 1, 4, 1, 0, 1, 5, 2, 0, 1, 6, 0, 0, 1, 7, 2, 1, 1, 9, 1, 0, 1, 10, 3, 0, 1, 12, 2, 0, 1, 14, 3, 1, 1, 1, 17, 1, 0, 0, 1, 19, 5, 1, 0, 1, 23, 3, 0, 0, 1, 26, 5, 2, 0, 1, 30, 4, 0, 0, 1, 35, 7, 0, 0

Offset

0,11

Links

Felix Huber, Table of n, a(n) for n = 0..10009 (rows 0..1095, flattened)

Formula

G.f.: Sum_{n >= 0} T(n, k) x^n = Product_{m >= 1, A001222(m) = k} 1/(1 - x^m).

T(n, 0) = 1. For n >= 1 the only allowed part is 1; for n = 0 this counts the empty partition.

T(n, 1) = A000607(n).

T(n, 2) = A101048(n).

T(n, 3) = A112313(n).

T(n, 4) = A112314(n).

T(n, 5) = A112315(n).

T(n, 6) = A112316(n).

Sum_{k >= 0} T(n, k) = A319169(n).

For n >= 1, T(n, floor(log_2(n))) = 1 iff Omega(n) = floor(log_2(n)); otherwise 0.

Example

The triangle begins:
  n\k |  0   1   2   3
  ----+---------------
   0  |  1
   1  |  1
   2  |  1   1
   3  |  1   1
   4  |  1   1   1
   5  |  1   2   0
   6  |  1   2   1
   7  |  1   3   0
   8  |  1   3   1   1
   9  |  1   4   1   0
  10  |  1   5   2   0
  11  |  1   6   0   0
  12  |  1   7   2   1
T(10, 2) = 2 since the two partitions [4, 6] and [10] both consist only of parts having exactly 2 prime factors (counted with multiplicity).

Maple

N := 22: # Enlarge if you want more rows
K := ilog2(N):
t := Array(0 .. N, 0 .. K):
for k from 0 to K do
    t[0, k] := 1
end do:
v := Vector(N):
for i from 1 to N do
    v[i] := NumberTheory:-Omega(i)
end do:
for k from 0 to K do
    for i from 1 to N do
        if v[i] = k then
            for n from i to N do
                t[n, k] := t[n, k] + t[n - i, k];
            end do
        end if
    end do
end do:
T := (n, k) -> t[n, k]:
seq(seq(T(n, k), k = 0 .. ilog2(n)), n = 0 .. N);

Crossrefs

Cf. A000607, A001222, A101048, A112313, A112314, A112315, A112316, A319169.

Sequence in context: A025853 A228247 A025847 * A367771 A334152 A092130

Adjacent sequences: A394208 A394209 A394210 * A394212 A394213 A394214

Keyword

nonn,tabf,easy

Author

Felix Huber, May 15 2026

Status

approved