OFFSET
1,3
COMMENTS
REFERENCES
Edmund Landau, Handbuch der Lehre von der Verteilung der Primzahlen, Chelsea Publishing, NY 1953, Vol. 1, p. 211, Eq. (5).
LINKS
Alois P. Heinz, Rows n = 1..10000, flattened
N. J. A. Sloane, The first 100 rows.
FORMULA
For fixed k, T(n,k) ~ (1/(k-1)!) * n * (log log n)^(k-1) / log n [Landau].
From Alois P. Heinz, Aug 19 2021: (Start)
Sum_{k>=1} k * T(n,k) = A013939(n).
Sum_{k>=1} k^2 * T(n,k) = A069811(n).
Sum_{k>=1} (-1)^(k-1) * T(n,k) = A123066(n).
Sum_{k>=1} (-1)^k * T(n,k) = -1 + A174863(n).
Sum_{k>=1} T(n,k) = n - 1. (End)
EXAMPLE
Rows 1 through 12 are:
1 [0]
2 [1]
3 [2]
4 [3]
5 [4]
6 [4, 1]
7 [5, 1]
8 [6, 1]
9 [7, 1]
10 [7, 2]
11 [8, 2]
12 [8, 3]
13 [9, 3]
MAPLE
omega := proc(n) nops(numtheory[factorset](n)) end proc: # # A001221
A:=Array(1..20, 0);
ans:=[[0]];
mx:=0;
for n from 2 to 100 do
k:=omega(n);
if k>mx then mx:=k; fi;
A[k]:=A[k]+1;
ans:=[op(ans), [seq(A[i], i=1..mx)]];
od:
ans;
# second Maple program:
b:= proc(n) option remember; `if`(n=0, 0,
b(n-1)+x^nops(ifactors(n)[2]))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..max(1, degree(p))))(b(n)):
seq(T(n), n=1..40); # Alois P. Heinz, Aug 19 2021
MATHEMATICA
T[n_] := If[n == 1, {0},
Range[n] // PrimeNu // Tally // Rest // #[[All, 2]]&];
Array[T, 40] // Flatten (* Jean-François Alcover, Mar 08 2022 *)
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
N. J. A. Sloane, Aug 19 2021
STATUS
approved