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A146290 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.

%I A146290 #10 Feb 16 2025 08:33:09
%S A146290 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,
%T A146290 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,
%U A146290 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
%N A146290 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.
%C A146290 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).
%C A146290 Every row that appears in A146289 appears exactly once in the table. Rows appear in order of first appearance in A146289.
%C A146290 T(n,0)=1.
%H A146290 Anonymous?, <a href="http://xrjunque.nom.es/precis/polycalc.aspx">Polynomial calculator</a>
%H A146290 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DistinctPrimeFactors.html">Distinct Prime Factors</a>
%H A146290 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 A146290 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).
%e A146290 Rows begin:
%e A146290   1;
%e A146290   1,1;
%e A146290   1,2;
%e A146290   1,2,1;
%e A146290   1,3;
%e A146290   1,3,2;
%e A146290   1,4;
%e A146290   1,4,3;...
%e A146290 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).
%Y A146290 For the number of distinct prime factors of n, see A001221.
%Y A146290 Row sums equal A146288(n). T(n, 1)=A036041(n) for n>1. T(n, A061394(n))=A052306(n).
%Y A146290 Row A098719(n) of this table is identical to row n of A007318.
%Y A146290 Cf. A146289. Also cf. A146291, A146292.
%K A146290 nonn,tabf
%O A146290 1,5
%A A146290 _Matthew Vandermast_, Nov 11 2008