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

1,5

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

Links

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

Example

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

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 *)

Prog

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

Crossrefs

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

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

Sequence in context: A228107 A140207 A104763 * A181322 A004070 A180562

Adjacent sequences: A027748 A027749 A027750 * A027752 A027753 A027754

Keyword

nonn,easy,tabf

Author

N. J. A. Sloane.

Extensions

More terms from Patrick De Geest, May 15 1998

Example edited by Omar E. Pol, Nov 23 2010

Status

approved