A146290 - OEIS

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

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OFFSET

1,5

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.

LINKS

Table of n, a(n) for n=1..102.

Anonymous?, Polynomial calculator

Eric Weisstein's World of Mathematics, Distinct Prime Factors

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

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).

EXAMPLE

Rows begin:

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).

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.

Sequence in context: A188884 A116679 A350032 * A347045 A323345 A135539

Adjacent sequences: A146287 A146288 A146289 * A146291 A146292 A146293

KEYWORD

nonn,tabf

AUTHOR

Matthew Vandermast, Nov 11 2008

STATUS

approved