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A333437 Triangle read by rows: T(n,k) is the number of Egyptian fractions 1 = 1/x_1 + ... + 1/x_k , with 0 < x_1 <= ... <= x_k = n.

%I A333437 #35 Mar 24 2020 22:30:25
%S A333437 1,0,1,0,0,1,0,0,1,1,0,0,0,0,1,0,0,1,3,2,1,0,0,0,0,0,0,1,0,0,0,1,3,3,
%T A333437 2,1,0,0,0,0,2,2,3,2,1,0,0,0,1,3,6,7,5,3,1,0,0,0,0,0,0,0,0,0,0,1,0,0,
%U A333437 0,3,8,15,21,24,20,11,4,1
%N A333437 Triangle read by rows: T(n,k) is the number of Egyptian fractions 1 = 1/x_1 + ... + 1/x_k , with 0 < x_1 <= ... <= x_k = n.
%H A333437 <a href="/index/Ed#Egypt">Index entries for sequences related to Egyptian fractions</a>
%F A333437 T(n,n) = 1.
%F A333437 If n is prime, T(n,k) = 0 for 1 <= k < n.
%e A333437 1 = 1/2 + 1/6 + 1/6 + 1/6 = 1/3 + 1/3 + 1/6 + 1/6 = 1/3 + 1/4 + 1/4 + 1/6. So T(6,4) = 3.
%e A333437 Triangle begins:
%e A333437 n\k  | 1  2  3  4  5   6   7   8   9  10 11 12
%e A333437 -----+----------------------------------------
%e A333437    1 | 1;
%e A333437    2 | 0, 1;
%e A333437    3 | 0, 0, 1;
%e A333437    4 | 0, 0, 1, 1;
%e A333437    5 | 0, 0, 0, 0, 1;
%e A333437    6 | 0, 0, 1, 3, 2,  1;
%e A333437    7 | 0, 0, 0, 0, 0,  0,  1;
%e A333437    8 | 0, 0, 0, 1, 3,  3,  2,  1;
%e A333437    9 | 0, 0, 0, 0, 2,  2,  3,  2,  1;
%e A333437   10 | 0, 0, 0, 1, 3,  6,  7,  5,  3,  1;
%e A333437   11 | 0, 0, 0, 0, 0,  0,  0,  0,  0,  0, 1;
%e A333437   12 | 0, 0, 0, 3, 8, 15, 21, 24, 20, 11, 4, 1;
%Y A333437 Row sums give A092666.
%Y A333437 Cf. A020473, A333496.
%K A333437 nonn,tabl
%O A333437 1,19
%A A333437 _Seiichi Manyama_, Mar 24 2020