A182469 Triangle read by rows in which row n lists the odd divisors of n.

1, 1, 1, 3, 1, 1, 5, 1, 3, 1, 7, 1, 1, 3, 9, 1, 5, 1, 11, 1, 3, 1, 13, 1, 7, 1, 3, 5, 15, 1, 1, 17, 1, 3, 9, 1, 19, 1, 5, 1, 3, 7, 21, 1, 11, 1, 23, 1, 3, 1, 5, 25, 1, 13, 1, 3, 9, 27, 1, 7, 1, 29, 1, 3, 5, 15, 1, 31, 1, 1, 3, 11, 33, 1, 17, 1, 5, 7, 35, 1

Offset: 1

Author

Reinhard Zumkeller, Apr 30 2012

Keywords

• nonn
• tabf

Comments

n-th row = intersection of A005408 and of n-th row of A027750.

Examples

			The triangle begins:
.  1   {1}
.  2   {1}
.  3   {1,3}
.  4   {1}
.  5   {1,5}
.  6   {1,3}
.  7   {1,7}
.  8   {1}
.  9   {1,3,9}
. 10   {1,5}
. 11   {1,11}
. 12   {1,3}
. 13   {1,13}
. 14   {1,7}
. 15   {1,3,5,15}
. 16   {1} .
		

Links

• Reinhard Zumkeller, Rows n = 1..2500 of triangle, flattened

Crossrefs

Cf. A001227 (row lengths), A000593 (row sums), A136655 (row products).

Cf. A000265, A005408, A027750, A050999, A051000, A037283, A037284, A037285, A171565.

Cf. also A237048.

Programs

Haskell

a182469 n k = a182469_tabf !! (n-1) !! (k-1)
a182469_row = a027750_row . a000265
a182469_tabf = map a182469_row [1..]

Mathematica

Flatten[Table[Select[Divisors[n],OddQ],{n,40}]] (* Harvey P. Dale, Aug 13 2012 *)
Flatten[Table[Divisors[n / 2^IntegerExponent[n, 2]], {n, 40}]] (* Amiram Eldar, May 02 2025 *)

PARI

tabf(nn) = {for (n=1, nn, fordiv(n, d, if (d%2, print1(d, ", "))); print(););} \\ Michel Marcus, Apr 22 2017

PARI

row(n) = divisors(n >> valuation(n, 2)); \\ Amiram Eldar, May 02 2025

Python

from sympy import divisors
def row(n):
    return [d for d in divisors(n) if d % 2]
for n in range(1, 21): print(row(n)) # Indranil Ghosh, Apr 22 2017

Formula

T(n,k) = A027750(A000265(n),k), 1 <= k <= A001227(n).

A000265(n) = T(n,A001227(n)).