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

1,3

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

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

Links

Franklin T. Adams-Watters, Rows 1..1000, flattened

Franklin T. Adams-Watters, Rows 1..10000

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

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

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

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.

Sequence in context: A348135 A368194 A233773 * A275055 A254679 A343651

Adjacent sequences: A027747 A027748 A027749 * A027751 A027752 A027753

Keyword

nonn,easy,tabf,look

Author

N. J. A. Sloane

Extensions

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

Status

approved