A146292 Triangle T(n,m) read by rows (n >= 1, 0 <= m <= A036041(n)), giving the number of divisors of A025487(n) with m prime factors (counted with multiplicity).

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 2, 1, 1, 2, 2, 2, 2, 1, 1, 3, 4, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 3, 2, 1, 1, 2, 2, 2, 2, 2, 1, 1, 3, 4, 4, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 3, 3, 2, 1, 1, 3, 5, 5, 3, 1, 1, 2, 2, 2, 2

Offset: 1

Author

Matthew Vandermast, Nov 11 2008

Keywords

• nonn
• tabf

Comments

All rows are palindromic. T(n, 0) = T(n, A036041(n)) = 1.

Every row that appears in A146291 appears exactly once in the table. Rows appear in order of first appearance in A146291.

Examples

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

Links

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

Crossrefs

For the number of prime factors of n counted with multiplicity, see A001222.

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

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

Cf. A146291. Also cf. A146289, A146290.

Formula

If A025487(n)'s canonical factorization 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).