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A146290
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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.
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10
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,5
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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.
(n,0)=1.
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LINKS
| Anonymous?, Polynomial calculator
Eric Weisstein's World of Mathematics, Distinct Prime Factors
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 Product p^e(p), then T(n, m) is the coefficient of k^m in the polynomial expansion of Product_p (1 + ek).
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EXAMPLE
| Rows begin: 1; 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).
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CROSSREFS
| For the number of distinct prime factors of n, see A001221.
Row sums equal A146288(n). (n, 1)=A036041(n) for n>1. (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: A008289 A188884 A116679 * A135539 A129264 A135840
Adjacent sequences: A146287 A146288 A146289 * A146291 A146292 A146293
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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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