A146290 Triangle T(n,m) read by rows (n >= 1, 0 <= m <= A061394(n)), giving the number of divisors of A025487(n) with m distinct prime factors.

1, 1, 1, 1, 2, 1, 2, 1, 1, 3, 1, 3, 2, 1, 4, 1, 4, 3, 1, 3, 3, 1, 1, 5, 1, 4, 4, 1, 5, 4, 1, 4, 5, 2, 1, 6, 1, 5, 6, 1, 6, 5, 1, 5, 7, 3, 1, 7, 1, 6, 8, 1, 5, 8, 4, 1, 7, 6, 1, 4, 6, 4, 1, 1, 6, 9, 1, 6, 9, 4, 1, 8, 1, 7, 10, 1, 6, 11, 6, 1, 8, 7, 1, 5, 9, 7, 2, 1, 7, 12, 1, 7, 11, 5, 1, 9, 1, 8, 12, 1, 7, 14

Offset: 1

Author

Matthew Vandermast, Nov 11 2008

Keywords

• nonn
• tabf

Comments

The formula used in obtaining the A025487(n)th row (see below) also gives the number of divisors of the k-th power of A025487(n).

Every row that appears in A146289 appears exactly once in the table. Rows appear in order of first appearance in A146289.

T(n,0)=1.

Examples

			Rows begin:
  1;
  1,1;
  1,2;
  1,2,1;
  1,3;
  1,3,2;
  1,4;
  1,4,3;...
36's 9 divisors include 1 divisor with 0 distinct prime factors (1); 4 with 1 (2, 3, 4 and 9); and 4 with 2 (6, 12, 18 and 36). Since 36 = A025487(11), the 11th row of the table therefore reads (1, 4, 4). These are the positive coefficients of the polynomial equation 1 + 4k + 4k^2 = (1 + 2k)(1 + 2k), derived from the prime factorization of 36 (namely, 2^2*3^2).
		

Links

• Anonymous?, Polynomial calculator
• Eric Weisstein's World of Mathematics, Distinct Prime Factors
• G. Xiao, WIMS server, Factoris (both expands and factors polynomials)

Crossrefs

For the number of distinct prime factors of n, see A001221.

Row sums equal A146288(n). T(n, 1)=A036041(n) for n>1. T(n, A061394(n))=A052306(n).

Row A098719(n) of this table is identical to row n of A007318.

Cf. A146289. Also cf. A146291, A146292.

Formula

If A025487(n)'s canonical factorization 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 + ek).