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

Author

Antti Karttunen, Jun 20 2000

Keywords

• nonn
• tabf

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

Examples

			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;
		

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

Crossrefs

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

Programs

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

Magma

[Reverse(Divisors(n)) : n in [1..30]];

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 *)

PARI

row(n)=Vecrev(divisors(n)) \\ Charles R Greathouse IV, Sep 02 2015

Formula

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

Extensions

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