A292226 -id:A292226 - cp's OEIS frontend

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

N. J. A. Sloane

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

			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

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

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.

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]];
```

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

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

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

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

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

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

A291127 Consider the zeros of the polynomial P(m,x) whose coefficients are the divisors of a number m. The sequence lists the numbers m such that P(m,x) contains at least two zeros that are purely imaginary numbers.

Original entry on oeis.org

6, 8, 10, 14, 15, 21, 22, 26, 27, 33, 34, 35, 38, 39, 42, 46, 51, 54, 55, 57, 58, 62, 65, 66, 69, 74, 77, 78, 82, 85, 86, 87, 88, 91, 93, 94, 95, 102, 104, 106, 110, 111, 114, 115, 118, 119, 122, 123, 125, 128, 129, 130, 133, 134, 136, 138, 141, 142, 143, 145

Offset: 1

Michel Lagneau, Aug 18 2017

nonn

P(m,x) = Sum_{i=1..k} d(i)*x^(i-1) where d(1), d(2), ..., d(k) are the k divisors of m.
The number of zeros of the polynomial P(n,x) is given by A032741(n).
We observe that all the zeros of the polynomial are located in the unit circle.
Conjecture: A032741(a(n))= p where p == 3 (mod 4), p prime. Examples:
A032741(m) = 3 for m = 6, 8, 10, 14, 15, 21, 22, 26, 27, 33, ...
A032741(m) = 7 for m = 42, 54, 66, 78, 88, ...
A032741(m) = 11 for m = 156, 204, 228, 276, 294, 342, 348, 372, ...
A032741(m) = 19 for m = 2544, 2832, 2928, 3216, 3408, 3504, 3792, ...
A032741(m) = 23 for m = 24492, 25428, 26052, 26364, 26988, 27924, ...
Except for the two purely imaginary zeros of P(m,x), it seems that the complex zeros are of the form u +- u*i where u is a real number.
From Wolfdieter Lang, Nov 07 2017: (Start)
P(m,x) is the row m polynomial of A027750 with increasing powers of x.
The numbers m = 1 and m = prime obviously do not appear in this sequence. The composite numbers m belonging to irreducible polynomials P(m,x) over the integers given in A292226 also do not appear in this sequence. Moreover, the composite numbers m with factorizable P(m,x) without a factor of the type a*x^2 + b, with positive integers a and b, also do not appear in this sequence; these are the numbers 18, 20, 28, 32, 44, ... (End)
Are there numbers m with more than one pair of purely imaginary solutions? - Wolfdieter Lang, Nov 14 2017
From Robert Israel, Nov 14 2017: (Start)
The even and odd parts of P(m,x) are of the form A(x^2) and x*B(x^2) for polynomials A and B with integer coefficients, and pairs of imaginary roots of P(m,x) correspond to negative roots of the gcd of A and B.
Includes the following:
  p^k where p is prime and k==3 (mod 4).
  p*q^k where k is odd and p, q are prime with either p < q or p > q^k.
  p*q*r^k where p, q, r are distinct primes and r > p*q.
(End)

			42 is in the sequence because P(42,x) = 1 + 2x + 3x^2 + 6x^3 + 7x^4 + 14x^5 + 21x^6 + 42x^7 = (1 + 2*x)*(1 + 3*x^2)*(1 + 7*x^4), and the seven zeros are -1/2, +(1/3)*sqrt(3)*i, -(1/3)*sqrt(3)*i, r*(1+i), r*(1-i), r*(-1+i), r*(-1-i) with r = 7^(3/4)*sqrt(2)/14. The relevant factor for the two purely imaginary zeros is (1 + 3*x^2). - _Wolfdieter Lang_, Nov 13 2017

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A027750, A032741, A292226.

F:= proc(n) local x,d,i,A,B,R;
  d:= sort(convert(numtheory:-divisors(n),list));
  A:= add(d[2*i]*x^(i-1),i=1..nops(d)/2);
  B:= add(d[2*i+1]*x^i,i=0..(nops(d)-1)/2);
  R:= gcd(A,B);
  sturm(sturmseq(R,x),x,-infinity,0) > 0;
end proc:
select(F, [$1..1000]); # Robert Israel, Nov 14 2017

Position[#, k_ /; k >= 2][[All, 1]] &@ Table[Count[Re /@ Values@ Apply[Join, Solve[Normal@ SeriesData[x, 0, #, 0, Length@ #, 1] == 0, x]], 0] &@ Divisors@ n, {n, 150}] (* Michael De Vlieger, Aug 21 2017 *)

isok(n) = {my(d = divisors(n), p = sum(k=1, #d, x^(k-1)*d[k])); #select(x->(real(x)==0), polroots(p)) >= 2;} \\ Michel Marcus, Sep 09 2017

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

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