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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.
5

%I #29 Jan 02 2021 17:13:09

%S 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,

%T 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,

%U 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

%N 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.

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

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

%H Reinhard Zumkeller, <a href="/A243987/b243987.txt">Rows n = 1..125 of triangle, flattened</a>

%H Dennis P. Walsh, <a href="http://capone.mtsu.edu/dwalsh/FACTORNK.pdf">Notes on counting the divisors of n</a>

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

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

%F T(n,k) = Sum_{j=1..k} A051731(n,j). - _Reinhard Zumkeller_, Apr 22 2015

%e T(6,4)=3 since there are 3 divisors of 6 that are less than or equal to 4, namely, 1, 2 and 3.

%e T(n,k) as a triangle, n=1..15:

%e 1,

%e 1, 2,

%e 1, 1, 2,

%e 1, 2, 2, 3,

%e 1, 1, 1, 1, 2,

%e 1, 2, 3, 3, 3, 4,

%e 1, 1, 1, 1, 1, 1, 2,

%e 1, 2, 2, 3, 3, 3, 3, 4,

%e 1, 1, 2, 2, 2, 2, 2, 2, 3,

%e 1, 2, 2, 2, 3, 3, 3, 3, 3, 4

%e 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2,

%e 1, 2, 3, 4, 4, 5, 5, 5, 5, 5, 5, 6,

%e 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2,

%e 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 4,

%e 1, 1, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4

%p T:=(n,k)->1/n!*eval(diff(sum(x^j/(1-x^j),j=1..k),x$n),x=0):

%p seq(seq(T(n,k), k=1..n), n=1..10);

%p # Alternative:

%p IversonBrackets := expr -> subs(true=1, false=0, evalb(expr)):

%p T := (n, k) -> add(IversonBrackets(irem(n, j) = 0), j = 1..k):

%p for n from 1 to 19 do seq(T(n, k), k = 1..n) od; # _Peter Luschny_, Jan 02 2021

%o (PARI) T(n, k) = sumdiv(n, d, d<=k); \\ _Michel Marcus_, Jun 17 2014

%o (Haskell)

%o a243987 n k = a243987_tabl !! (n-1) !! (k-1)

%o a243987_row n = a243987_tabl !! (n-1)

%o a243987_tabl = map (scanl1 (+)) a051731_tabl

%o -- _Reinhard Zumkeller_, Apr 22 2015

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

%Y Cf. A138553, A051731.

%Y Cf. A340260, A340261.

%K nonn,tabl

%O 1,3

%A _Dennis P. Walsh_, Jun 16 2014