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A146291
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Triangle T(n,m) read by rows (n >= 1, 0 <= m <= A001222(n)), giving the number of divisors of n with m prime factors (counted with multiplicity).
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4
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1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,12
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COMMENTS
| All rows are palindromic. (n,0)=(n, A001222(n))=1.
Two numbers have identical rows in the table if and only if they have the same prime signature.
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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 the canonical factorization of n 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; 1,1,1; 1,1; 1,2,1; 1,1; 1,1,1,1; 1,1,1; 1,2,1;...
12 has 1 divisor with 0 total prime factors (1), 2 with 1 (2 and 3), 2 with 2 (4 and 6) and 1 with 3 (12), for a total of 6. The 12th row of the table therefore reads (1, 2, 2, 1). These are the positive coefficients of the polynomial 1 + 2k + 2k^2 + (1)k^3 = (1 + k + k^2)(1 + k), derived from the prime factorization of 12 (namely, 2^2*3^1).
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CROSSREFS
| Row sums equal A000005(n). (n, 1)= A001221(n) for n>1.
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. A146292. Also cf. A146289, A146290.
Sequence in context: A172098 A204162 A043285 * A086251 A177121 A092931
Adjacent sequences: A146288 A146289 A146290 * A146292 A146293 A146294
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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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