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A346617
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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.
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9
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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, 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, 12, 1, 18, 12, 1, 18, 13, 1, 18, 14, 1, 18, 15, 1, 18, 16, 1, 19, 16, 1, 19, 17, 1
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OFFSET
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1,3
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COMMENTS
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Column k >= 1 of the triangle gives the number of numbers i in the range 1 <= i <= n with omega(i) = A001221(i) = k.
A285577 is a similar triangle which has an extra column on the left for k = 0.
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REFERENCES
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Edmund Landau, Handbuch der Lehre von der Verteilung der Primzahlen, Chelsea Publishing, NY 1953, Vol. 1, p. 211, Eq. (5).
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LINKS
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FORMULA
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For fixed k, T(n,k) ~ (1/(k-1)!) * n * (log log n)^(k-1) / log n [Landau].
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)
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EXAMPLE
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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]
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MAPLE
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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)):
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MATHEMATICA
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T[n_] := If[n == 1, {0},
Range[n] // PrimeNu // Tally // Rest // #[[All, 2]]&];
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CROSSREFS
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Row lengths give A111972 (for n>1).
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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