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A197871
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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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0
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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, 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, 56, 38, 21, 9, 4, 1, 44, 62, 48, 24, 11, 4, 2, 48, 75, 51, 29, 13, 6, 2
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OFFSET
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1,2
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REFERENCES
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G. J. O. Jameson, The Prime Number Theorem, Cambridge, 2004, p.145.
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LINKS
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EXAMPLE
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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.
The triangle begins:
0
2 1
4 3 1
6 6 2 1
9 9 4 2
11 13 7 3 1
15 17 10 4 2
...
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MATHEMATICA
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Join[{0}, Flatten[Table[Transpose[Tally[Table[Plus @@ Last /@ FactorInteger[i], {i, 2, n^2}]]][[2]], {n, 2, 15}]]]
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PROG
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(PARI) T(n, k) = #select(x->(bigomega(x) == k), [1..n^2]);
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
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CROSSREFS
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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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