cp's OEIS Frontend

A146292 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).

%I A146292 #9 Feb 16 2025 08:33:09
%S A146292 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,
%T A146292 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,
%U A146292 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
%N A146292 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).
%C A146292 All rows are palindromic. T(n, 0) = T(n, A036041(n)) = 1.
%C A146292 Every row that appears in A146291 appears exactly once in the table. Rows appear in order of first appearance in A146291.
%H A146292 Anonymous?, <a href="http://xrjunque.nom.es/precis/polycalc.aspx">Polynomial calculator</a>
%H A146292 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Roundness.html">Roundness</a>
%H A146292 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 A146292 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).
%e A146292 Rows begin:
%e A146292   1;
%e A146292   1,1;
%e A146292   1,1,1;
%e A146292   1,2,1;
%e A146292   1,1,1,1;
%e A146292   1,2,2,1;
%e A146292   1,1,1,1,1;...
%e A146292 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).
%Y A146292 For the number of prime factors of n counted with multiplicity, see A001222.
%Y A146292 Row sums equal A146288(n). T(n, 1) = A061394(n) for n>1.
%Y A146292 Row A098719(n) of this table is identical to row n of A007318.
%Y A146292 Cf. A146291. Also cf. A146289, A146290.
%K A146292 nonn,tabf
%O A146292 1,8
%A A146292 _Matthew Vandermast_, Nov 11 2008