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A340261 T(n, k) is the number of integers that are less than or equal to k that do not divide n. Triangle read by rows, for 0 <= k <= n.

%I A340261 #10 Jan 03 2021 04:34:36
%S A340261 0,0,0,0,1,1,0,0,1,1,0,1,2,3,3,0,0,0,1,2,2,0,1,2,3,4,5,5,0,0,1,1,2,3,
%T A340261 4,4,0,1,1,2,3,4,5,6,6,0,0,1,2,2,3,4,5,6,6,0,1,2,3,4,5,6,7,8,9,9,0,0,
%U A340261 0,0,1,1,2,3,4,5,6,6,0,1,2,3,4,5,6,7,8,9,10,11,11
%N A340261 T(n, k) is the number of integers that are less than or equal to k that do not divide n. Triangle read by rows, for 0 <= k <= n.
%F A340261 T(n, k) = Sum_{j=1..k} [n mod j <> 0], where [ ] are the Iverson brackets.
%F A340261 T(n, k) = card({j : j = 1..k} \ divisors(n)).
%e A340261 Table starts:
%e A340261                             [1] 0;
%e A340261                            [2] 0, 0;
%e A340261                           [3] 0, 1, 1;
%e A340261                         [4] 0, 0, 1, 1;
%e A340261                        [5] 0, 1, 2, 3, 3;
%e A340261                      [6] 0, 0, 0, 1, 2, 2;
%e A340261                     [7] 0, 1, 2, 3, 4, 5, 5;
%e A340261                   [8] 0, 0, 1, 1, 2, 3, 4, 4;
%e A340261                  [9] 0, 1, 1, 2, 3, 4, 5, 6, 6;
%e A340261                [10] 0, 0, 1, 2, 2, 3, 4, 5, 6, 6;
%p A340261 IversonBrackets := expr -> subs(true=1, false=0, evalb(expr)):
%p A340261 T := (n, k) -> add(IversonBrackets(irem(n, j) <> 0), j = 1..k):
%p A340261 # Alternative:
%p A340261 T := (n, k) -> nops({seq(j, j = 1..k)} minus numtheory:-divisors(n)):
%p A340261 for n from 1 to 19 do seq(T(n, k), k = 1..n) od;
%Y A340261 T(n, n) = n - tau(n) = A049820(n).
%Y A340261 T(2*n, n) = n + 1 - tau(2*n) = A234306(n).
%Y A340261 Cf. A000005, A051731, A243987, A340260.
%K A340261 nonn,tabl
%O A340261 1,13
%A A340261 _Peter Luschny_, Jan 02 2021