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A089789 Number of irreducible factors of Gauss polynomials.

%I A089789 #14 Mar 24 2017 00:47:50
%S A089789 0,0,0,0,1,0,0,1,1,0,0,2,2,2,0,0,1,2,2,1,0,0,3,3,4,3,3,0,0,1,3,3,3,3,
%T A089789 1,0,0,3,3,5,4,5,3,3,0,0,2,4,4,5,5,4,4,2,0,0,3,4,6,5,7,5,6,4,3,0,0,1,
%U A089789 3,4,5,5,5,5,4,3,1,0,0,5,5,7,7,9,7,9,7,7,5,5,0,0,1,5,5,6,7,7,7,7,6,5,5,1,0
%N A089789 Number of irreducible factors of Gauss polynomials.
%C A089789 T(n,k) is the number of irreducible factors of the (separable) polynomial [n]!/([k]![n-k]!). Here [n]! denotes the product of the first n quantum integers, the n-th quantum integer being defined as (1-q^n)/(1-q).
%C A089789 T(n,k) gives the number of positive integers m <= n such that (n mod m) < (k mod m). - _Tom Edgar_, Aug 21 2014
%H A089789 Stan Wagon and Herbert S. Wilf, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v1i1r3">When are subset sums equidistributed modulo m?</a>, The Electronic Journal of Combinatorics, Vol. 1, 1994 (#R3).
%F A089789 T(n, k) = T(n-1, k-1) + d(n) - d(k), where d(n) is the number of divisors of n.
%e A089789 The triangle T(n,k) begins:
%e A089789 n\k  0  1  2  3  4  5  6  7  8  9  10  11  12  13 ...
%e A089789 0:   0
%e A089789 1:   0  0
%e A089789 2:   0  1  0
%e A089789 3:   0  1  1  0
%e A089789 4:   0  2  2  2  0
%e A089789 5:   0  1  2  2  1  0
%e A089789 6:   0  3  3  4  3  3  0
%e A089789 7:   0  1  3  3  3  3  1  0
%e A089789 8:   0  3  3  5  4  5  3  3  0
%e A089789 9:   0  2  4  4  5  5  4  4  2  0
%e A089789 10:  0  3  4  6  5  7  5  6  4  3   0
%e A089789 11:  0  1  3  4  5  5  5  5  4  3   1   0
%e A089789 12:  0  5  5  7  7  9  7  9  7  7   5   5   0
%e A089789 13:  0  1  5  5  6  7  7  7  7  6   5   5   1   0
%e A089789 ... Formatted by _Wolfdieter Lang_, Dec 07 2012
%e A089789 T(8,3) equals the number of irreducible factors of (1-q^8)(1-q^7)(1-q^6)/((1-q^3)(1-q^2)(1-q)), which is a product of 5 cyclotomic polynomials in q, namely the 2nd, 4th, 6th, 7th and 8th. Thus T(8,3)=5.
%Y A089789 Cf. A008967, A047971, A219237, A000005.
%K A089789 easy,nonn,tabl
%O A089789 0,12
%A A089789 _Paul Boddington_, Jan 09 2004