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.

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, 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, 0, 0, 1, 1, 2, 3, 4, 5, 6, 6, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 11

Offset: 1

Author

Peter Luschny, Jan 02 2021

Keywords

• nonn
• tabl

Examples

			Table starts:
                            [1] 0;
                           [2] 0, 0;
                          [3] 0, 1, 1;
                        [4] 0, 0, 1, 1;
                       [5] 0, 1, 2, 3, 3;
                     [6] 0, 0, 0, 1, 2, 2;
                    [7] 0, 1, 2, 3, 4, 5, 5;
                  [8] 0, 0, 1, 1, 2, 3, 4, 4;
                 [9] 0, 1, 1, 2, 3, 4, 5, 6, 6;
               [10] 0, 0, 1, 2, 2, 3, 4, 5, 6, 6;
		

Crossrefs

T(n, n) = n - tau(n) = A049820(n).

T(2*n, n) = n + 1 - tau(2*n) = A234306(n).

Cf. A000005, A051731, A243987, A340260.

Programs

Maple

IversonBrackets := expr -> subs(true=1, false=0, evalb(expr)):
T := (n, k) -> add(IversonBrackets(irem(n, j) <> 0), j = 1..k):
# Alternative:
T := (n, k) -> nops({seq(j, j = 1..k)} minus numtheory:-divisors(n)):
for n from 1 to 19 do seq(T(n, k), k = 1..n) od;

Formula

T(n, k) = Sum_{j=1..k} [n mod j <> 0], where [ ] are the Iverson brackets.

T(n, k) = card({j : j = 1..k} \ divisors(n)).