1,7

The formula used in obtaining the nth row (see below) also gives the number of divisors of the kth power of n.

Two numbers have identical rows in the table if and only if they have the same prime signature.

(n,0)=1.

Anonymous?, Polynomial calculator

Eric Weisstein's World of Mathematics, Distinct Prime Factors

G. Xiao, WIMS server, Factoris (both expands and factors polynomials)

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 + ek).

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).

Row sums equal A000005(n). (n, 1)=A001222(n) for n>1. (n, (A01221(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.

Sequence in context: A059233 A143898 A101873 this_sequence A079211 A081418 A088951

Adjacent sequences: A146286 A146287 A146288 this_sequence A146290 A146291 A146292

nonn,tabf

Matthew Vandermast (ghodges14(AT)comcast.net), Nov 11 2008