%I #10 Oct 02 2024 14:30:29
%S 1,1,1,1,1,1,1,1,2,3,2,1,1,1,2,2,1,1,1,3,5,5,5,3,1,1,1,3,5,5,3,1,1,1,
%T 4,7,7,7,7,7,4,1,1,3,6,7,7,7,7,6,3,1,1,5,9,8,7,9,7,8,9,5,1,1,1,5,5,10,
%U 7,7,10,5,5,1,1,1,6,11,11,11,11,11,11,11,11,11,6,1
%N Irregular triangle read by rows: T(n,k) is the largest divisor of binomial(n,k) which is less than n, with 0 <= k <= n.
%D R. K. Guy, Unsolved Problems in Number Theory, Springer, 1st edition, 1981. See section B23.
%H Stefano Spezia, <a href="/A376611/b376611.txt">Table of n, a(n) for n = 2..5251</a> (first 100 rows of the triangle)
%e The irregular triangle begins as:
%e 1, 1, 1;
%e 1, 1, 1, 1;
%e 1, 2, 3, 2, 1;
%e 1, 1, 2, 2, 1, 1;
%e 1, 3, 5, 5, 5, 3, 1;
%e 1, 1, 3, 5, 5, 3, 1, 1;
%e 1, 4, 7, 7, 7, 7, 7, 4, 1;
%e 1, 3, 6, 7, 7, 7, 7, 6, 3, 1;
%e 1, 5, 9, 8, 7, 9, 7, 8, 9, 5, 1;
%e 1, 1, 5, 5, 10, 7, 7, 10, 5, 5, 1, 1;
%e ...
%t T[n_,k_]:=Max[Select[Divisors[Binomial[n,k]],#<n &]];Table[T[n,k],{n,2,12},{k,0,n}]//Flatten
%Y Cf. A000012 (n=k or k=0), A007318, A027750, A032742 (k=1).
%Y Cf. A376612 (less than k).
%K nonn,look,tabf
%O 2,9
%A _Stefano Spezia_, Sep 29 2024