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A146291 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).

%I A146291 #23 Feb 16 2025 08:33:09
%S A146291 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,
%T A146291 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,
%U A146291 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
%N A146291 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).
%C A146291 All rows are palindromic. T(n,0) = T(n,A001222(n)) = 1.
%C A146291 Two numbers have identical rows in the table if and only if they have the same prime signature.
%C A146291 If n is a perfect square then Sum_{even m} T(n,m) = 1 + Sum_{odd m} T(n,m), otherwise Sum_{even m} T(n,m) = Sum_{odd m} T(n,m). - _Geoffrey Critzer_, Feb 08 2015
%H A146291 Alois P. Heinz, <a href="/A146291/b146291.txt">Rows n = 1..2500, flattened</a>
%H A146291 Anonymous?, <a href="http://xrjunque.nom.es/precis/polycalc.aspx">Polynomial calculator</a>
%H A146291 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Roundness.html">Roundness</a>
%H A146291 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 A146291 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).
%e A146291 Rows begin:
%e A146291 1;
%e A146291 1, 1;
%e A146291 1, 1;
%e A146291 1, 1, 1;
%e A146291 1, 1;
%e A146291 1, 2, 1;
%e A146291 1, 1;
%e A146291 1, 1, 1, 1;
%e A146291 1, 1, 1;
%e A146291 1, 2, 1;
%e A146291 ...
%e A146291 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).
%p A146291 with(numtheory):
%p A146291 T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(
%p A146291          add(x^bigomega(d), d=divisors(n))):
%p A146291 seq(T(n), n=1..100);  # _Alois P. Heinz_, Feb 25 2015
%t A146291 Join[{{1}},
%t A146291 Table[nn = DivisorSigma[0, n];
%t A146291   CoefficientList[
%t A146291    Series[Product[(1 - x^i)/(1 - x), {i,
%t A146291 FactorInteger[n][[All, 2]] + 1}], {x, 0, nn}], x], {n, 2, 100}]] (* _Geoffrey Critzer_, Jan 01 2015 *)
%Y A146291 Row sums equal A000005(n). T(n,1) = A001221(n) for n>1.
%Y A146291 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 A146291 Cf. A146292, A146289, A146290.
%K A146291 nonn,tabf
%O A146291 1,12
%A A146291 _Matthew Vandermast_, Nov 11 2008