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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.
2
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
1,13
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)).
EXAMPLE
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;
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;
CROSSREFS
T(n, n) = n - tau(n) = A049820(n).
T(2*n, n) = n + 1 - tau(2*n) = A234306(n).
Sequence in context: A265590 A260208 A242457 * A190146 A026932 A087401
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, Jan 02 2021
STATUS
approved