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A001013
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Jordan-Polya numbers: products of factorial numbers A000142.
(Formerly M0993 N0372)
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55
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1, 2, 4, 6, 8, 12, 16, 24, 32, 36, 48, 64, 72, 96, 120, 128, 144, 192, 216, 240, 256, 288, 384, 432, 480, 512, 576, 720, 768, 864, 960, 1024, 1152, 1296, 1440, 1536, 1728, 1920, 2048, 2304, 2592, 2880, 3072, 3456, 3840, 4096, 4320, 4608, 5040, 5184, 5760
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OFFSET
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1,2
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COMMENTS
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Also, numbers of the form 1^d_1*2^d_2*3^d_3*...*k^d_k where k, d_1, ..., d_k are natural numbers satisfying d_1 >= d_2 >= d_3 >= ... >= d_k >= 1. - N. J. A. Sloane, Jun 14 2015
Possible orders of automorphism groups of trees.
Except for the numbers 2, 9 and 10 this sequence is conjectured to be the same as A034878.
Equivalently, (a(n)/6)*(6*x^2 - 6*x + (6*x-3)*a(n) + 2*a(n)^2 + 1) = N^2 has an integer solution. - Ralf Stephan, Dec 04 2004
Named after the French mathematician Camille Jordan (1838-1922) and the Hungarian mathematician George Pólya (1887-1985). - Amiram Eldar, May 22 2021
Possible numbers of transitive orientations of comparability graphs (Golumbic, 1977). - David Eppstein, Dec 29 2021
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REFERENCES
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Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B23, p. 123.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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EXAMPLE
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864 = (3!)^2*4!.
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MAPLE
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N:= 10000: # get all terms <= N
S:= {1}:
for k from 2 do
kf:= k!;
if kf > N then break fi;
S := S union {seq(seq(kf^j * s, j = 1 .. floor(log[kf](N/s))), s=S)};
od:
S; # if using Maple 11 or earlier, uncomment the next line:
# sort(convert(S, list));
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MATHEMATICA
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For[p=0; a=f=Table[n!, {n, 1, 8}], p=!=a, p=a; a=Select[Union@@Outer[Times, f, a], #<=8!&]]; a
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PROG
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(Sage) # uses[prod_hull from A246663]
(Haskell)
import Data.Set (empty, fromList, deleteFindMin, union)
import qualified Data.Set as Set (null)
a001013 n = a001013_list !! (n-1)
a001013_list = 1 : h 0 empty [1] (drop 2 a000142_list) where
h z s mcs xs'@(x:xs)
| Set.null s || x < m = h z (union s (fromList $ map (* x) mcs)) mcs xs
| m == z = h m s' mcs xs'
| otherwise = m : h m (union s' (fromList (map (* m) $ init (m:mcs)))) (m:mcs) xs'
where (m, s') = deleteFindMin s
(PARI) list(lim, mx=lim)=if(lim<2, return([1])); my(v=[1], t=1); for(n=2, mx, t*=n; if(t>lim, break); v=concat(v, t*list(lim\t, t))); Set(v) \\ Charles R Greathouse IV, May 18 2015
(Python)
def aupto(lim, mx=None):
if lim < 2: return [1]
v, t = [1], 1
if mx == None: mx = lim
for k in range(2, mx+1):
t *= k
if t > lim: break
v += [t*rest for rest in aupto(lim//t, t)]
return sorted(set(v))
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CROSSREFS
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KEYWORD
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nonn,nice,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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