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A146291
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Triangle T(n,m) read by rows (n >= 1, 0 <= m <= A001222(n)), giving the number of divisors of n with m prime factors (counted with multiplicity).
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24
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1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2
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OFFSET
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1,12
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COMMENTS
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All rows are palindromic. T(n,0) = T(n,A001222(n)) = 1.
Two numbers have identical rows in the table if and only if they have the same prime signature.
If n is a perfect square then Sum_{even m} T(n,m) = 1 + Sum_{odd m} T(n,m), otherwise Sum_{even m} T(n,m) = Sum_{odd m} T(n,m). - Geoffrey Critzer, Feb 08 2015
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LINKS
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Eric Weisstein's World of Mathematics, Roundness
G. Xiao, WIMS server, Factoris (both expands and factors polynomials)
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FORMULA
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If the canonical factorization of n into prime powers is the product of p^e(p), then T(n,m) is the coefficient of k^m in the polynomial expansion of Product_p (sum_{i=0..e} k^i).
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EXAMPLE
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Rows begin:
1;
1, 1;
1, 1;
1, 1, 1;
1, 1;
1, 2, 1;
1, 1;
1, 1, 1, 1;
1, 1, 1;
1, 2, 1;
...
12 has 1 divisor with 0 total prime factors (1), 2 with 1 (2 and 3), 2 with 2 (4 and 6) and 1 with 3 (12), for a total of 6. The 12th row of the table therefore reads (1, 2, 2, 1). These are the positive coefficients of the polynomial 1 + 2k + 2k^2 + (1)k^3 = (1 + k + k^2)(1 + k), derived from the prime factorization of 12 (namely, 2^2*3^1).
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MAPLE
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with(numtheory):
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(
add(x^bigomega(d), d=divisors(n))):
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MATHEMATICA
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Join[{{1}},
Table[nn = DivisorSigma[0, n];
CoefficientList[
Series[Product[(1 - x^i)/(1 - x), {i,
FactorInteger[n][[All, 2]] + 1}], {x, 0, nn}], x], {n, 2, 100}]] (* Geoffrey Critzer, Jan 01 2015 *)
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CROSSREFS
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Row n of A007318 is identical to row A002110(n) of this table and also identical to the row for any squarefree number with n prime factors.
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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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