A146289 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.

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

Offset: 1

Author

Matthew Vandermast, Nov 11 2008

Keywords

• nonn
• tabf

Comments

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.

Examples

			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).
		

Links

• G. C. Greubel, Table of n, a(n) for the first 500 rows, flattened
• Anonymous?, Polynomial calculator
• Eric Weisstein's World of Mathematics, Distinct Prime Factors
• G. Xiao, WIMS server, Factoris (both expands and factors polynomials)

Crossrefs

Row sums equal A000005(n).

T(n, 1) = A001222(n) for n>1. T(n, A001221(n)) = A005361(n).

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.

Cf. A146290. Also cf. A146291, A146292.

Programs

Maple

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:
seq(f(n),n=1..100); # Robert Israel, Feb 10 2015

Mathematica

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 *)

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

Formula

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).