Search: id:A146289
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%I A146289
%S A146289 1,1,1,1,1,1,2,1,1,1,2,1,1,1,1,3,1,2,1,2,1,1,1,1,3,2,1,1,1,2,1,1,2,1,1,
%T A146289 4,1,1,1,3,2,1,1,1,3,2,1,2,1,1,2,1,1,1,1,4,3,1,2,1,2,1,1,3,1,3,2,1,1,1,
%U A146289 3,3,1,1,1,1,5,1,2,1,1,2,1,1,2,1,1,4,4,1,1,1,2,1,1,2,1,1,4,3,1,1,1,3,3
%N A146289 Triangle T(n,m) read by rows (n >= 1, 0 <= m <= A001221(n)), giving the 
               number of divisors of n with m distinct prime factors.
%C A146289 The formula used in obtaining the nth row (see below) also gives the 
               number of divisors of the kth power of n.
%C A146289 Two numbers have identical rows in the table if and only if they have 
               the same prime signature.
%C A146289 (n,0)=1.
%H A146289 Anonymous?, <a href="http://xrjunque.nom.es/precis/polycalc.aspx">Polynomial 
               calculator</a>
%H A146289 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               DistinctPrimeFactors.html">Distinct Prime Factors</a>
%H A146289 G. Xiao, WIMS server, <a href="http://wims.unice.fr/~wims/en_tool~algebra~factor.en.html">
               Factoris</a> (both expands and factors polynomials)
%F A146289 If the canonical factorization of n 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).
%e A146289 Rows begin: 1; 1,1; 1,1; 1,2; 1,1; 1,2,1; 1,1; 1,3; 1,2; 1,2,1;...
%e A146289 12 has 1 divisor with 0 distinct prime factors (1); 3 with 1 (2, 3 and 
               4); and 2 with 2 (6 and 12), for a total of 6. The 12th row of the 
               table therefore reads (1, 3, 2). These are the positive coefficients 
               of the polynomial equation 1 + 3k + 2k^2 = (1 + 2k)(1 + k), derived 
               from the prime factorization of 12 (namely, 2^2*3^1).
%Y A146289 Row sums equal A000005(n). (n, 1)=A001222(n) for n>1. (n, (A01221(n))=A005361(n).
%Y A146289 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.
%Y A146289 Cf. A146290. Also cf. A146291, A146292.
%Y A146289 Sequence in context: A059233 A143898 A101873 this_sequence A079211 A081418 
               A088951
%Y A146289 Adjacent sequences: A146286 A146287 A146288 this_sequence A146290 A146291 
               A146292
%K A146289 nonn,tabf
%O A146289 1,7
%A A146289 Matthew Vandermast (ghodges14(AT)comcast.net), Nov 11 2008

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