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A146292
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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).
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6
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,8
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COMMENTS
| All rows are palindromic. (n, 0) = (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.
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LINKS
| Anonymous?, Polynomial calculator
Eric Weisstein's World of Mathematics, Roundness
G. Xiao, WIMS server, Factoris (both expands and factors polynomials)
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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).
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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).
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CROSSREFS
| For the number of prime factors of n counted with multiplicity, see A001222.
Row sums equal A146288(n). (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: A037830 A174353 A187447 * A139039 A122172 A025910
Adjacent sequences: A146289 A146290 A146291 * A146293 A146294 A146295
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KEYWORD
| nonn,tabf
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AUTHOR
| Matthew Vandermast (ghodges14(AT)comcast.net), Nov 11 2008
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