A173455 -id:A173455 - cp's OEIS frontend

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

N. J. A. Sloane

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

			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

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

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

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

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

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

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

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

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

More terms from Patrick De Geest, May 15 1998

Example edited by Omar E. Pol, Nov 23 2010

A091818 Sum of the even proper divisors of 2n. Sum of the even divisors of 2n that are less than 2n.

Original entry on oeis.org

0, 2, 2, 6, 2, 12, 2, 14, 8, 16, 2, 32, 2, 20, 18, 30, 2, 42, 2, 44, 22, 28, 2, 72, 12, 32, 26, 56, 2, 84, 2, 62, 30, 40, 26, 110, 2, 44, 34, 100, 2, 108, 2, 80, 66, 52, 2, 152, 16, 86, 42, 92, 2, 132, 34, 128, 46, 64, 2, 216, 2, 68, 82, 126, 38, 156, 2, 116

Offset: 1

Mohammad K. Azarian, Mar 07 2004

			The sum of the even divisors of 18 that are less than 18 is 8 = 2+6.

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A001065, A032741, A091570, A074400, A173455.

Table[Total[Select[Most[Divisors[2 n]],EvenQ]],{n,70}] (* Harvey P. Dale, Apr 28 2023 *)

a(n) = sumdiv(2*n, d, !(d%2) * d * (d<2*n)); \\ Michel Marcus, Jan 14 2014

from sympy import divisors
def a(n): return sum(d for d in divisors(2*n) if d%2==0) - 2*n
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Oct 30 2017

a(n) = A074400(2n) - 2n. - Michel Marcus, Jan 14 2014
a(n) = Sum_{d|2n, d<2n, d even} d. - Wesley Ivan Hurt, Mar 02 2022
a(n) = 2 * A001065(n). - Alois P. Heinz, Mar 02 2022

More terms from Michel Marcus, Jan 14 2014

cp's OEIS Frontend

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.

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