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A162247
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Irregular triangle in which row n lists all factorizations of n, sorted by the number of factors in each factorization.
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68
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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
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,2
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COMMENTS
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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.
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REFERENCES
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LINKS
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EXAMPLE
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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;
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MATHEMATICA
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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}]]
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PROG
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(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]
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CROSSREFS
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KEYWORD
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nice,tabf,nonn
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AUTHOR
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STATUS
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approved
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