A243987 Triangle read by rows: T(n, k) is the number of divisors of n that are less than or equal to k for 1 <= k <= n.

1, 1, 2, 1, 1, 2, 1, 2, 2, 3, 1, 1, 1, 1, 2, 1, 2, 3, 3, 3, 4, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 3, 3, 3, 3, 4, 1, 1, 2, 2, 2, 2, 2, 2, 3, 1, 2, 2, 2, 3, 3, 3, 3, 3, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 3, 4, 4, 5, 5, 5, 5, 5, 5, 6, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 4, 1, 1, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4

Offset

1,3

Comments

This triangular sequence T(n,k) generalizes sequence A000005, the number of divisors of n; in particular, A000005(n) = T(n,n).

Also, for prime p, T(p,k) = 1 when k < p and T(p,p) = 2.

Links

Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened

Dennis P. Walsh, Notes on counting the divisors of n

Formula

T(n,1) = 1; T(n,n) = A000005(n).

T(n,k) = coefficient of the x^n term in the expansion of Sum(x^j/(1-x^j), j=1..k).

T(n,k) = Sum_{j=1..k} A051731(n,j). - Reinhard Zumkeller, Apr 22 2015

Example

T(6,4)=3 since there are 3 divisors of 6 that are less than or equal to 4, namely, 1, 2 and 3.
T(n,k) as a triangle, n=1..15:
1,
1, 2,
1, 1, 2,
1, 2, 2, 3,
1, 1, 1, 1, 2,
1, 2, 3, 3, 3, 4,
1, 1, 1, 1, 1, 1, 2,
1, 2, 2, 3, 3, 3, 3, 4,
1, 1, 2, 2, 2, 2, 2, 2, 3,
1, 2, 2, 2, 3, 3, 3, 3, 3, 4
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2,
1, 2, 3, 4, 4, 5, 5, 5, 5, 5, 5, 6,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2,
1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 4,
1, 1, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4

Maple

T:=(n, k)->1/n!*eval(diff(sum(x^j/(1-x^j), j=1..k), x$n), x=0):
seq(seq(T(n, k), k=1..n), n=1..10);
# Alternative:
IversonBrackets := expr -> subs(true=1, false=0, evalb(expr)):
T := (n, k) -> add(IversonBrackets(irem(n, j) = 0), j = 1..k):
for n from 1 to 19 do seq(T(n, k), k = 1..n) od; # Peter Luschny, Jan 02 2021

Prog

(PARI) T(n, k) = sumdiv(n, d, d<=k); \\ Michel Marcus, Jun 17 2014
(Haskell)
a243987 n k = a243987_tabl !! (n-1) !! (k-1)
a243987_row n = a243987_tabl !! (n-1)
a243987_tabl = map (scanl1 (+)) a051731_tabl
-- Reinhard Zumkeller, Apr 22 2015

Crossrefs

Cf. A000005 (diagonal), A000012 (first column), A081307 (row sums), A027750 (divisors of n).

Cf. A138553, A051731.

Cf. A340260, A340261.

Sequence in context: A116674 A025836 A029319 * A050205 A281530 A340260

Adjacent sequences: A243984 A243985 A243986 * A243988 A243989 A243990

Keyword

nonn,tabl

Author

Dennis P. Walsh, Jun 16 2014

Status

approved