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Irregular triangle T(n,k) of the number of numbers with k prime factors (repetitions allowed) less than n^2.
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%I #17 Aug 16 2022 16:05:30

%S 0,2,1,4,3,1,6,6,2,1,9,9,4,2,11,13,7,3,1,15,17,10,4,2,18,22,13,7,2,1,

%T 22,26,19,8,4,1,25,34,22,12,4,2,30,40,28,13,7,2,34,48,32,18,7,3,1,39,

%U 56,38,21,9,4,1,44,62,48,24,11,4,2,48,75,51,29,13,6,2

%N Irregular triangle T(n,k) of the number of numbers with k prime factors (repetitions allowed) less than n^2.

%D G. J. O. Jameson, The Prime Number Theorem, Cambridge, 2004, p.145.

%e In the third row, reading from the left, 6 is the number of primes <= 16, 6 is the number of semiprimes <= 16, 2 is the number of numbers with three prime divisors (repetitions allowed) <= 16, and 1 is the number of numbers with four divisors <= 16.

%e The triangle begins:

%e 0

%e 2 1

%e 4 3 1

%e 6 6 2 1

%e 9 9 4 2

%e 11 13 7 3 1

%e 15 17 10 4 2

%e ...

%t Join[{0}, Flatten[Table[Transpose[Tally[Table[Plus @@ Last /@ FactorInteger[i], {i, 2, n^2}]]][[2]], {n, 2, 15}]]]

%o (PARI) T(n,k) = #select(x->(bigomega(x) == k), [1..n^2]);

%o row(n) = my(v = vector(n, k, T(n,k))); my(pos); for (k=1, n, if (v[k], pos=k)); Vec(v, pos); \\ _Michel Marcus_, Aug 16 2022

%Y Similar to A052130.

%K nonn,tabf

%O 1,2

%A _Daniel Tisdale_, Oct 18 2011