A056538 Irregular triangle read by rows: row n lists the divisors of n in decreasing order.

1, 2, 1, 3, 1, 4, 2, 1, 5, 1, 6, 3, 2, 1, 7, 1, 8, 4, 2, 1, 9, 3, 1, 10, 5, 2, 1, 11, 1, 12, 6, 4, 3, 2, 1, 13, 1, 14, 7, 2, 1, 15, 5, 3, 1, 16, 8, 4, 2, 1, 17, 1, 18, 9, 6, 3, 2, 1, 19, 1, 20, 10, 5, 4, 2, 1, 21, 7, 3, 1, 22, 11, 2, 1, 23, 1, 24, 12, 8, 6, 4, 3, 2, 1, 25, 5, 1, 26, 13, 2, 1, 27, 9

Offset

1,2

Comments

Old name was "Replace n by its divisors in reverse order."

This gives the second elements of the ordered pairs (a,b), a >= 1, b >= 1, ordered by their product ab.

T(n,k) = n / A027750(n,k) = A027750(n,n-k+1), 1 <= k <= A000005(n). - Reinhard Zumkeller, Sep 28 2014

The 2nd column of the triangle is the largest proper divisor (A032742). - Charles Kusniec, Jan 30 2021

Links

Reinhard Zumkeller, Rows n = 1..1000 of triangle, flattened

Omar E. Pol, Illustration of the divisors of n - Omar E. Pol, Nov 22 2009

Formula

a(n) = A064894(A064896(n)).

Example

Triangle begins:
1;
2, 1;
3, 1;
4, 2, 1;
5, 1;
6, 3, 2, 1;
7, 1;
8, 4, 2, 1;
9, 3, 1;
10, 5, 2, 1;
11, 1;
12, 6, 4, 3, 2, 1;
13, 1;
14, 7, 2, 1;
15, 5, 3, 1;
16, 8, 4, 2, 1;
17, 1;
18, 9, 6, 3, 2, 1;
19, 1;
20, 10, 5, 4, 2, 1;

Maple

map(op, [seq(reverse(sort(divisors(j))), j=1..30)]);
cdr := proc(l) if 0 = nops(l) then ([]) else (l[2..nops(l)]): fi: end:
reverse := proc(l) if 0 = nops(l) then ([]) else [op(reverse(cdr(l))), l[1]]; fi: end:

Mathematica

Table[Reverse@ Divisors@ n, {n, 27}] // Flatten (* Michael De Vlieger, Jul 27 2016 *)

Prog

(Magma) [Reverse(Divisors(n)) : n in [1..30]];
(Haskell)
a056538 n k = a056538_tabf !! (n-1) !! (k-1)
a056538_row n = a056538_tabf !! (n-1)
a056538_tabf = map reverse a027750_tabf
-- Reinhard Zumkeller, Sep 28 2014
(PARI) row(n)=Vecrev(divisors(n)) \\ Charles R Greathouse IV, Sep 02 2015

Crossrefs

Cf. A027750 for the first elements, A056534, A168017, A000005 (row lengths), A000203 (row sums), A032742 (largest proper divisor).

Sequence in context: A319848 A364749 A233772 * A266742 A120385 A216477

Adjacent sequences: A056535 A056536 A056537 * A056539 A056540 A056541

Keyword

nonn,tabf

Author

Antti Karttunen, Jun 20, 2000

Extensions

Definition revised by N. J. A. Sloane, Jul 27 2016

Status

approved