OFFSET
1,3
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
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
Read as a linear sequence, this is generated by the D'Hondt (or Jefferson) apportionment method applied to the weights p(k) = 1/k, choosing the smallest candidate number in case of ties. (The D'Hondt method can be applied even though the sum of the weights is infinite.) In other words, a(n) is the smallest k minimizing k*(c(k)+1), where c(k) is the number of occurrences of k among the previous terms. Furthermore, a(n)*(c(a(n))+1) equals the index of the row in which a(n) appears. - Pontus von Brömssen, Mar 29 2026
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, D'Hondt method.
Wikipedia, Table of divisors.
FORMULA
T(n,k) = n / A056538(n,k) = A056538(n,n-k+1), 1 <= k <= A000005(n). - Reinhard Zumkeller, Sep 28 2014
EXAMPLE
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
MAPLE
seq(op(numtheory:-divisors(a)), a = 1 .. 20); # Matt C. Anderson, May 15 2017
MATHEMATICA
Flatten[ Table[ Flatten [ Divisors[ n ] ], {n, 1, 30} ] ]
PROG
(Magma) [Divisors(n) : n in [1..20]];
(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
(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
CROSSREFS
KEYWORD
AUTHOR
EXTENSIONS
More terms from Scott Lindhurst (ScottL(AT)alumni.princeton.edu)
STATUS
approved