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A346617 Irregular triangle T(n,m) read by rows (n >= 1, 1 <= m <= Max(A001221([1..n]))): T(n,m) = number of integers in [1,n] with m distinct prime factors.

%I A346617 #40 May 03 2025 04:03:56
%S A346617 0,1,2,3,4,4,1,5,1,6,1,7,1,7,2,8,2,8,3,9,3,9,4,9,5,10,5,11,5,11,6,12,
%T A346617 6,12,7,12,8,12,9,13,9,13,10,14,10,14,11,15,11,15,12,16,12,16,12,1,17,
%U A346617 12,1,18,12,1,18,13,1,18,14,1,18,15,1,18,16,1,19,16,1,19,17,1
%N A346617 Irregular triangle T(n,m) read by rows (n >= 1, 1 <= m <= Max(A001221([1..n]))): T(n,m) = number of integers in [1,n] with m distinct prime factors.
%C A346617 Column k >= 1 of the triangle gives the number of numbers i in the range 1 <= i <= n with omega(i) = A001221(i) = k.
%C A346617 A285577 is a similar triangle which has an extra column on the left for k = 0.
%D A346617 G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, Cambridge, University Press, 1940, pp. 52-56.
%D A346617 Edmund Landau, Handbuch der Lehre von der Verteilung der Primzahlen, Chelsea Publishing, NY 1953, Vol. 1, p. 211, Eq. (5).
%H A346617 Alois P. Heinz, <a href="/A346617/b346617.txt">Rows n = 1..10000, flattened</a>
%H A346617 N. J. A. Sloane, <a href="/A346617/a346617.txt">The first 100 rows.</a>
%F A346617 For fixed k, T(n,k) ~ (1/(k-1)!) * n * (log log n)^(k-1) / log n [Landau].
%F A346617 From _Alois P. Heinz_, Aug 19 2021: (Start)
%F A346617 Sum_{k>=1} k * T(n,k) = A013939(n).
%F A346617 Sum_{k>=1} k^2 * T(n,k) = A069811(n).
%F A346617 Sum_{k>=1} (-1)^(k-1) * T(n,k) = A123066(n).
%F A346617 Sum_{k>=1} (-1)^k * T(n,k) = -1 + A174863(n).
%F A346617 Sum_{k>=1} T(n,k) = n - 1. (End)
%e A346617 Rows 1 through 12 are:
%e A346617 1 [0]
%e A346617 2 [1]
%e A346617 3 [2]
%e A346617 4 [3]
%e A346617 5 [4]
%e A346617 6 [4, 1]
%e A346617 7 [5, 1]
%e A346617 8 [6, 1]
%e A346617 9 [7, 1]
%e A346617 10 [7, 2]
%e A346617 11 [8, 2]
%e A346617 12 [8, 3]
%e A346617 13 [9, 3]
%p A346617 omega := proc(n) nops(numtheory[factorset](n)) end proc: # # A001221
%p A346617 A:=Array(1..20,0);
%p A346617 ans:=[[0]];
%p A346617 mx:=0;
%p A346617 for n from 2 to 100 do
%p A346617 k:=omega(n);
%p A346617 if k>mx then mx:=k; fi;
%p A346617 A[k]:=A[k]+1;
%p A346617 ans:=[op(ans),[seq(A[i],i=1..mx)]];
%p A346617 od:
%p A346617 ans;
%p A346617 # second Maple program:
%p A346617 b:= proc(n) option remember; `if`(n=0, 0,
%p A346617       b(n-1)+x^nops(ifactors(n)[2]))
%p A346617     end:
%p A346617 T:= n-> (p-> seq(coeff(p, x, i), i=1..max(1, degree(p))))(b(n)):
%p A346617 seq(T(n), n=1..40);  # _Alois P. Heinz_, Aug 19 2021
%t A346617 T[n_] := If[n == 1, {0},
%t A346617      Range[n] // PrimeNu // Tally // Rest // #[[All, 2]]&];
%t A346617 Array[T, 40] // Flatten (* _Jean-François Alcover_, Mar 08 2022 *)
%Y A346617 Cf. A001221, A013939, A069811, A123066, A174863, A285577.
%Y A346617 Row lengths give A111972 (for n>1).
%K A346617 nonn,tabf
%O A346617 1,3
%A A346617 _N. J. A. Sloane_, Aug 19 2021