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A001013 Jordan-Polya numbers: products of factorial numbers A000142.
(Formerly M0993 N0372)
55
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
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
REFERENCES
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).
LINKS
Reinhard Zumkeller, Table of n, a(n) for n = 1..10000 (first 987 terms from T. D. Noe)
Jean-Marie De Koninck, Nicolas Doyon, A. Arthur Bonkli Razafindrasoanaivolala and William Verreault, Bounds for the counting function of the Jordan-Pólya numbers, Archivum Mathematicum, Vol. 56, No. 3 (2020), pp. 141-152; also on arXiv, arXiv:2107.09114 [math.NT], 2021.
Martin Charles Golumbic, Comparability graphs and a new matroid, Journal of Combinatorial Theory, Series B, Vol. 22, No. 1 (1977), pp. 68-90.
Camille Jordan, Sur les assemblages de lignes, Journal für die reine und angewandte Mathematik, Vol. 70 (1869), pp. 185-190; alternative link.
Robert A. Melter, Autometrized unary algebras, J. Combinatorial Theory, Vol. 5, No. 1 (1968), pp. 21-29.
Eric Weisstein's World of Mathematics, Factorial Products.
EXAMPLE
864 = (3!)^2*4!.
MAPLE
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));
# Robert Israel, Sep 09 2014
MATHEMATICA
For[p=0; a=f=Table[n!, {n, 1, 8}], p=!=a, p=a; a=Select[Union@@Outer[Times, f, a], #<=8!&]]; a
PROG
(Sage) # uses[prod_hull from A246663]
prod_hull(factorial, 5760) # Peter Luschny, Sep 09 2014
(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
-- Reinhard Zumkeller, Nov 13 2014
(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))
print(aupto(5760)) # Michael S. Branicky, Jul 21 2021 after Charles R Greathouse IV
CROSSREFS
Cf. A000142, A034878, A093373 (complement), A344438 (characteristic function).
Union of A344181 and A344179. Subsequence of A025487 (see also A064783).
See also A359636 and A359751.
Sequence in context: A096850 A250270 A062847 * A115746 A025610 A344181
KEYWORD
nonn,nice,easy
AUTHOR
EXTENSIONS
More terms, formula from Christian G. Bower, Dec 15 1999
Edited by Dean Hickerson, Sep 17 2002
STATUS
approved

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Last modified April 16 23:10 EDT 2024. Contains 371755 sequences. (Running on oeis4.)