A027751 Irregular triangle read by rows in which row n lists the proper divisors of n (those divisors of n which are < n), with the first row {1} by convention.

1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 4, 1, 3, 1, 2, 5, 1, 1, 2, 3, 4, 6, 1, 1, 2, 7, 1, 3, 5, 1, 2, 4, 8, 1, 1, 2, 3, 6, 9, 1, 1, 2, 4, 5, 10, 1, 3, 7, 1, 2, 11, 1, 1, 2, 3, 4, 6, 8, 12, 1, 5, 1, 2, 13, 1, 3, 9, 1, 2, 4, 7, 14, 1, 1, 2, 3, 5, 6, 10, 15, 1, 1, 2, 4, 8, 16, 1, 3, 11, 1, 2, 17, 1, 5, 7, 1, 2, 3, 4, 6, 9, 12, 18

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• tabf

Comments

Or, take the list 1,2,3,4,... of natural numbers (A000027) and replace each number by its proper divisors.

The row length is 1 for n = 1 and A032741(n) for n >= 2. - Wolfdieter Lang, Jan 16 2016

Examples

			The irregular triangle T(n,k) begins:
n\k  1 2 3 4  5 ...
1:   1  (by convention)
2:   1
3:   1
4:   1 2
5:   1
6:   1 2 3
7:   1
8:   1 2 4
9:   1 3
10:  1 2 5
11:  1
12:  1 2 3 4  6
13:  1
14:  1 2 7
15:  1 3 5
16:  1 2 4 8
17:  1
18:  1 2 3 6  9
19:  1
20:  1 2 4 5 10
.... reformatted - _Wolfdieter Lang_, Jan 16 2016
		

Links

• Alois P. Heinz, Rows n = 1..1540, flattened

Crossrefs

Cf. A027750, A032741 (row lengths), A001065, A000005.

Row sums give A173455. - Omar E. Pol, Nov 23 2010

Programs

Haskell

a027751 n k = a027751_tabf !! (n-1) !! (k-1)
a027751_row n = a027751_tabf !! (n-1)
a027751_tabf = [1] : map init (tail a027750_tabf)
-- Reinhard Zumkeller, Apr 18 2012

Maple

with(numtheory):
T:= n-> sort([(divisors(n) minus {n})[]])[]: T(1):=1:
seq(T(n), n=1..50); # Alois P. Heinz, Apr 11 2012

Mathematica

Table[ Divisors[n] // Most, {n, 1, 36}] // Flatten // Prepend[#, 1] & (* Jean-François Alcover, Jun 10 2013 *)

PARI

row(n) = if (n==1, [1], my(d = divisors(n)); vector(#d-1,k, d[k])); \\ Michel Marcus, Apr 30 2017

Python

from sympy import divisors
def a(n): return [1] if n==1 else divisors(n)[:-1]
for n in range(21): print(a(n)) # Indranil Ghosh, Apr 30 2017

Extensions

More terms from Patrick De Geest, May 15 1998

Example edited by Omar E. Pol, Nov 23 2010