A027750 Triangle read by rows in which row n lists the divisors of n.

1, 1, 2, 1, 3, 1, 2, 4, 1, 5, 1, 2, 3, 6, 1, 7, 1, 2, 4, 8, 1, 3, 9, 1, 2, 5, 10, 1, 11, 1, 2, 3, 4, 6, 12, 1, 13, 1, 2, 7, 14, 1, 3, 5, 15, 1, 2, 4, 8, 16, 1, 17, 1, 2, 3, 6, 9, 18, 1, 19, 1, 2, 4, 5, 10, 20, 1, 3, 7, 21, 1, 2, 11, 22, 1, 23, 1, 2, 3, 4, 6, 8, 12, 24, 1, 5, 25, 1, 2, 13, 26, 1, 3, 9, 27, 1, 2, 4, 7, 14, 28, 1, 29

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• tabf
• look

Comments

Or, in the list of natural numbers (A000027), replace n with its divisors.

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

Also, row n lists the largest parts of the partitions of n whose parts are not distinct. - Omar E. Pol, Sep 17 2008

Concatenation of n-th row gives A037278(n). - Reinhard Zumkeller, Aug 07 2011

{A210208(n,k): k=1..A073093(n)} subset of {T(n,k): k=1..A000005(n)} for all n. - Reinhard Zumkeller, Mar 18 2012

Row sums give A000203. Right border gives A000027. - Omar E. Pol, Jul 29 2012

Indices of records are in A006218. - Irina Gerasimova, Feb 27 2013

The number of primes in the n-th row is omega(n) = A001221(n). - Michel Marcus, Oct 21 2015

The row polynomials P(n,x) = Sum_{k=1..A000005(n)} T(n,k)*x^k with composite n which are irreducible over the integers are given in A292226. - Wolfdieter Lang, Nov 09 2017

T(n,k) is also the number of parts in the k-th partition of n into equal parts (see example). - Omar E. Pol, Nov 20 2019

Let there be an infinite number of tiles, each labeled with a positive integer m, initially placed on square m of an infinite 1D board. At step n, the leftmost unblocked tile (i.e., the top tile of the leftmost nonempty stack) moves forward exactly m squares, where m is its label. Tiles that land on the same square form a stack, and only the top tile of any stack may move. This sequence records the label m of the tile that moves at step n. - Ali Sada, May 23 2025

All divisors of a positive integer n form a finite set. Extending divisibility to n = 0 by using the definition (k|n <=> exists m such that m*k = n) makes the set of divisors infinite, suggesting the definition was not intended for zero, as arithmetic functions typically apply to n >= 1. So to preserve a core property when generalizing (cardinality), one can define divisors of n >= 0 as the fixed points of the greatest common divisor on the set [n] = {0, 1, 2, ..., n}. By this definition, the divisors of 0 are {0}, since 0|0 and gcd(0, 0) = 0. This definition is not circular because the gcd can be effectively calculated using the Euclidean algorithm. (Cf. links.) - Peter Luschny, Jun 02 2025

Examples

			Triangle begins:
  1;
  1, 2;
  1, 3;
  1, 2, 4;
  1, 5;
  1, 2, 3, 6;
  1, 7;
  1, 2, 4, 8;
  1, 3, 9;
  1, 2, 5, 10;
  1, 11;
  1, 2, 3, 4, 6, 12;
  ...
For n = 6 the partitions of 6 into equal parts are [6], [3,3], [2,2,2], [1,1,1,1,1,1], so the number of parts are [1, 2, 3, 6] respectively, the same as the divisors of 6. - _Omar E. Pol_, Nov 20 2019
		

Links

• Franklin T. Adams-Watters, Rows 1..1000, flattened
• Franklin T. Adams-Watters, Rows 1..10000
• Peter Luschny, A Python program using gcd for n >= 0.
• Omar E. Pol, Illustration of initial terms, (2009).
• Eric Weisstein's World of Mathematics, Divisor
• Wikipedia, Table of divisors
• Index entries for sequences related to divisors of numbers

Crossrefs

Cf. A000005 (row length), A001221, A027749, A027751, A056534, A056538, A127093, A135010, A161700, A163280, A240698 (partial sums of rows), A240694 (partial products of rows), A247795 (parities), A292226, A244051.

Programs

Haskell

a027750 n k = a027750_row n !! (k-1)
a027750_row n = filter ((== 0) . (mod n)) [1..n]
a027750_tabf = map a027750_row [1..]
-- Reinhard Zumkeller, Jan 15 2011, Oct 21 2010

Magma

[Divisors(n) : n in [1..20]];

Maple

seq(op(numtheory:-divisors(a)), a = 1 .. 20) # Matt C. Anderson, May 15 2017

Mathematica

Flatten[ Table[ Flatten [ Divisors[ n ] ], {n, 1, 30} ] ]

PARI

v=List();for(n=1,20,fordiv(n,d,listput(v,d)));Vec(v) \\ Charles R Greathouse IV, Apr 28 2011

Python

from sympy import divisors
for n in range(1, 16):
    print(divisors(n)) # Indranil Ghosh, Mar 30 2017

Formula

a(A006218(n-1) + k) = k-divisor of n, 1 <= k <= A000005(n). - Reinhard Zumkeller, May 10 2006

T(n,k) = n / A056538(n,k) = A056538(n,n-k+1), 1 <= k <= A000005(n). - Reinhard Zumkeller, Sep 28 2014

Extensions

More terms from Scott Lindhurst (ScottL(AT)alumni.princeton.edu)