A146291 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).

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

Offset: 1

Author

Matthew Vandermast, Nov 11 2008

Keywords

• nonn
• tabf

Comments

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

Examples

			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).
		

Links

• Alois P. Heinz, Rows n = 1..2500, flattened
• Anonymous?, Polynomial calculator
• Eric Weisstein's World of Mathematics, Roundness
• G. Xiao, WIMS server, Factoris (both expands and factors polynomials)

Crossrefs

Row sums equal A000005(n). T(n,1) = A001221(n) for n>1.

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. A146292, A146289, A146290.

Programs

Maple

with(numtheory):
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(
         add(x^bigomega(d), d=divisors(n))):
seq(T(n), n=1..100);  # Alois P. Heinz, Feb 25 2015

Mathematica

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 *)

Formula

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).