A162247 Irregular triangle in which row n lists all factorizations of n, sorted by the number of factors in each factorization.

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

Offset

1,2

Comments

Row n begins with n because it is a factorization of length 1. In each factorization, the factors are in nondecreasing order. This sequence is A056472 with the factorizations in a different order. Sequence A001055(n) gives the number of factorizations of n; A066637(n) gives the number of numbers in row n. In the Mathematica program, the function f returns a list of the factorizations of n.

These factorizations are useful in determining the forms of numbers that have a given number of divisors. For example, to find the forms of numbers that have 12 divisors, we look at the four factorizations of 12 (12, 2*6, 3*4, 2*2*3), subtract 1 from each factor, and find the forms to be p^11, p q^5, p^2 q^3, and p q r^2, where p, q, and r are prime numbers.

References

See A001055.

Links

T. D. Noe, Rows n=1..1000 of triangle, flattened

R. J. Mathar, Factorizations of n=1..1100

Example

1;
2;
3;
4,2*2;
5;
6,2*3;
7;
8,2*4,2*2*2;
9,3*3;
10,2*5;
11;
12,2*6,3*4,2*2*3;

Mathematica

g[lst_, p_] := Module[{t, i, j}, Union[Flatten[Table[t=lst[[i]]; t[[j]]=p*t[[j]]; Sort[t], {i, Length[lst]}, {j, Length[lst[[i]]]}], 1], Table[Sort[Append[lst[[i]], p]], {i, Length[lst]}]]]; f[n_] := Module[{i, j, p, e, lst={{}}}, {p, e}=Transpose[FactorInteger[n]]; Do[lst=g[lst, p[[i]]], {i, Length[p]}, {j, e[[i]]}]; lst]; Flatten[Table[f[n], {n, 25}]]

Prog

(Haskell)
import Data.List (sortBy)
import Data.Ord (comparing)
a162247 n k = a162247_tabl !! (n-1) !! (k-1)
a162247_row n = a162247_tabl !! (n-1)
a162247_tabl = map (concat . sortBy (comparing length)) $ tail fss where
   fss = [] : map fact [1..] where
         fact x = [x] : [d : fs | d <- [2..x], let (x', r) = divMod x d,
                                  r == 0, fs <- fss !! x', d <= head fs]
-- Reinhard Zumkeller, Jan 08 2013

Crossrefs

Sequence in context: A182711 A138136 A056472 * A264809 A035578 A227784

Adjacent sequences: A162244 A162245 A162246 * A162248 A162249 A162250

Keyword

nice,tabf,nonn

Author

T. D. Noe, Jun 28 2009

Status

approved