A135539 Triangle read by rows: T(n,k) = number of divisors of n that are >= k.

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

Offset: 1

Author

Gary W. Adamson, Oct 30 2007

Keywords

• nonn
• easy
• tabl

Comments

Row sums give A000203.

Left border is A000005.

Examples

			First few rows of the triangle:
  1;
  2, 1;
  2, 1, 1;
  3, 2, 1, 1;
  2, 1, 1, 1, 1;
  4, 3, 2, 1, 1, 1;
  2, 1, 1, 1, 1, 1, 1;
  4, 3, 2, 2, 1, 1, 1, 1;
  3, 2, 2, 1, 1, 1, 1, 1, 1;
  4, 3, 2, 2, 2, 1, 1, 1, 1, 1;
  2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
  6, 5, 4, 3, 2, 2, 1, 1, 1, 1, 1, 1;
  ...
		

Links

• Seiichi Manyama, Rows n = 1..140, flattened

Crossrefs

Column k=1..10 give A000005, A032741, A023645, A321014, A338648, A338649, A338650, A338651, A338652, A338653.

Cf. A051731, A000203.

Cf. A001008, A001620, A002805.

Programs

Maple

with(numtheory);
f1:=proc(n) local d,s1,t1,t2,i;
d:=tau(n);
s1:=sort(divisors(n));
t1:=Array(1..n,0);
for i from 1 to d do t1[n-s1[i]+1]:=1; od:
t2:=PSUM(convert(t1,list));
[seq(t2[n+1-i],i=1..n)];
end proc;
for n from 1 to 15 do lprint(f1(n)); od: # N. J. A. Sloane, Nov 09 2018

Mathematica

T[n_, k_] := DivisorSum[n, Boole[# >= k]&];
Table[T[n, k], {n, 1, 15}, {k, 1, n}] // Flatten (* Jean-François Alcover, Feb 15 2023 *)

PARI

row(n) = my(d=divisors(n)); vector(n, k, #select(x->(x>=k), d)); \\ Michel Marcus, Jul 23 2022

Formula

Triangle read by rows, partial sums of A051731 starting from the right. A051731 as a lower triangular matrix times an all 1's lower triangular matrix.

From Seiichi Manyama, Jan 07 2023: (Start)

G.f. of column k: Sum_{j>=1} x^(k*j)/(1 - x^j).

G.f. of column k: Sum_{j>=k} x^j/(1 - x^j). (End)

Sum_{j=1..n} T(j, k) ~ n * (log(n) + 2*gamma - 1 - H(k-1)), where gamma is Euler's constant (A001620), and H(k) = A001008(k)/A002805(k) is the k-th harmonic number. - Amiram Eldar, Jan 08 2024

Extensions

Clearer definition from N. J. A. Sloane, Nov 09 2018