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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).
6
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
1,8
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.
LINKS
Eric Weisstein's World of Mathematics, Roundness
G. Xiao, WIMS server, Factoris (both expands and factors polynomials)
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).
EXAMPLE
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).
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.
Sequence in context: A174353 A319582 A187447 * A139039 A279061 A206491
KEYWORD
nonn,tabf
AUTHOR
Matthew Vandermast, Nov 11 2008
STATUS
approved