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A146289
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Triangle T(n,m) read by rows (n >= 1, 0 <= m <= A001221(n)), giving the number of divisors of n with m distinct prime factors.
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15
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1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 2, 1, 2, 1, 1, 1, 1, 3, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 4, 1, 1, 1, 3, 2, 1, 1, 1, 3, 2, 1, 2, 1, 1, 2, 1, 1, 1, 1, 4, 3, 1, 2, 1, 2, 1, 1, 3, 1, 3, 2, 1, 1, 1, 3, 3, 1, 1, 1, 1, 5, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 4, 4, 1, 1, 1, 2, 1, 1, 2, 1, 1, 4, 3, 1, 1, 1, 3, 3
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OFFSET
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1,7
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COMMENTS
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The formula used in obtaining the n-th row (see below) also gives the number of divisors of the k-th power of n.
Two numbers have identical rows in the table if and only if they have the same prime signature.
T(n,0)=1.
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LINKS
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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 Product p^e(p), then T(n, m) is the coefficient of k^m in the polynomial expansion of Product_p (1 + e(p) k).
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EXAMPLE
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Rows begin:
1;
1,1;
1,1;
1,2;
1,1;
1,2,1;
1,1;
1,3;
1,2;
1,2,1;
...
12 has 1 divisor with 0 distinct prime factors (1); 3 with 1 (2, 3 and 4); and 2 with 2 (6 and 12), for a total of 6. The 12th row of the table therefore reads (1, 3, 2). These are the positive coefficients of the polynomial equation 1 + 3k + 2k^2 = (1 + 2k)(1 + k), derived from the prime factorization of 12 (namely, 2^2*3^1).
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MAPLE
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f:= proc(n)
local F, G, f, t, k;
F:= ifactors(n)[2];
G:= mul(1+f[2]*t, f= F);
seq(coeff(G, t, k), k=0..nops(F));
end proc:
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MATHEMATICA
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Join[{{1}}, Table[nn = DivisorSigma[0, n]; CoefficientList[Series[Product[1 + i x, {i, FactorInteger[n][[All, 2]]}], {x, 0, nn}], x], {n, 2, 100}]] // Grid (* Geoffrey Critzer, Feb 09 2015 *)
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PROG
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(PARI) tabf(nn) = {for (n=1, nn, vd = divisors(n); vo = vector(#vd, k, omega(vd[k])); for (k=0, vecmax(vo), print1(#select(x->x==k, vo), ", "); ); print(); ); } \\ Michel Marcus, Apr 22 2017
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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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