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%I M0993 N0372 #110 Oct 27 2023 18:03:23
%S 1,2,4,6,8,12,16,24,32,36,48,64,72,96,120,128,144,192,216,240,256,288,
%T 384,432,480,512,576,720,768,864,960,1024,1152,1296,1440,1536,1728,
%U 1920,2048,2304,2592,2880,3072,3456,3840,4096,4320,4608,5040,5184,5760
%N Jordan-Polya numbers: products of factorial numbers A000142.
%C 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
%C Possible orders of automorphism groups of trees.
%C Except for the numbers 2, 9 and 10 this sequence is conjectured to be the same as A034878.
%C 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
%C Named after the French mathematician Camille Jordan (1838-1922) and the Hungarian mathematician George Pólya (1887-1985). - _Amiram Eldar_, May 22 2021
%C Possible numbers of transitive orientations of comparability graphs (Golumbic, 1977). - _David Eppstein_, Dec 29 2021
%D Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B23, p. 123.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Reinhard Zumkeller, <a href="/A001013/b001013.txt">Table of n, a(n) for n = 1..10000</a> (first 987 terms from T. D. Noe)
%H Jean-Marie De Koninck, Nicolas Doyon, A. Arthur Bonkli Razafindrasoanaivolala and William Verreault, <a href="https://doi.org/10.5817/AM2020-3-141">Bounds for the counting function of the Jordan-Pólya numbers</a>, Archivum Mathematicum, Vol. 56, No. 3 (2020), pp. 141-152; also <a href="https://arxiv.org/abs/2107.09114">on arXiv</a>, arXiv:2107.09114 [math.NT], 2021.
%H Martin Charles Golumbic, <a href="https://doi.org/10.1016/0095-8956(77)90049-1">Comparability graphs and a new matroid</a>, Journal of Combinatorial Theory, Series B, Vol. 22, No. 1 (1977), pp. 68-90.
%H Camille Jordan, <a href="https://doi.org/10.1515/crll.1869.70.185">Sur les assemblages de lignes</a>, Journal für die reine und angewandte Mathematik, Vol. 70 (1869), pp. 185-190; <a href="https://eudml.org/doc/148084">alternative link</a>.
%H Robert A. Melter, <a href="http://dx.doi.org/10.1016/S0021-9800(68)80025-0">Autometrized unary algebras</a>, J. Combinatorial Theory, Vol. 5, No. 1 (1968), pp. 21-29.
%H George Pólya, <a href="https://doi.org/10.1007/BF02546665">Kombinatorische Anzahlbestimmungen für Gruppen, Graphen und chemische Verbindungen</a>, Acta Mathematica, Vol. 68 (1937), pp. 145-254; <a href="http://archive.ymsc.tsinghua.edu.cn/pacm_download/117/5571-11511_2006_Article_BF02546665.pdf">alternative link</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/FactorialProducts.html">Factorial Products</a>.
%H <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>
%H <a href="/index/Tra#trees">Index entries for sequences related to trees</a>
%e 864 = (3!)^2*4!.
%p N:= 10000: # get all terms <= N
%p S:= {1}:
%p for k from 2 do
%p kf:= k!;
%p if kf > N then break fi;
%p S := S union {seq(seq(kf^j * s, j = 1 .. floor(log[kf](N/s))),s=S)};
%p od:
%p S; # if using Maple 11 or earlier, uncomment the next line:
%p # sort(convert(S,list));
%p # _Robert Israel_, Sep 09 2014
%t For[p=0; a=f=Table[n!, {n, 1, 8}], p=!=a, p=a; a=Select[Union@@Outer[Times, f, a], #<=8!&]]; a
%o (Sage) # uses[prod_hull from A246663]
%o prod_hull(factorial, 5760) # _Peter Luschny_, Sep 09 2014
%o (Haskell)
%o import Data.Set (empty, fromList, deleteFindMin, union)
%o import qualified Data.Set as Set (null)
%o a001013 n = a001013_list !! (n-1)
%o a001013_list = 1 : h 0 empty [1] (drop 2 a000142_list) where
%o h z s mcs xs'@(x:xs)
%o | Set.null s || x < m = h z (union s (fromList $ map (* x) mcs)) mcs xs
%o | m == z = h m s' mcs xs'
%o | otherwise = m : h m (union s' (fromList (map (* m) $ init (m:mcs)))) (m:mcs) xs'
%o where (m, s') = deleteFindMin s
%o -- _Reinhard Zumkeller_, Nov 13 2014
%o (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
%o (Python)
%o def aupto(lim, mx=None):
%o if lim < 2: return [1]
%o v, t = [1], 1
%o if mx == None: mx = lim
%o for k in range(2, mx+1):
%o t *= k
%o if t > lim: break
%o v += [t*rest for rest in aupto(lim//t, t)]
%o return sorted(set(v))
%o print(aupto(5760)) # _Michael S. Branicky_, Jul 21 2021 after _Charles R Greathouse IV_
%Y Cf. A000142, A034878, A093373 (complement), A344438 (characteristic function).
%Y Union of A344181 and A344179. Subsequence of A025487 (see also A064783).
%Y See also A359636 and A359751.
%K nonn,nice,easy
%O 1,2
%A _N. J. A. Sloane_
%E More terms, formula from _Christian G. Bower_, Dec 15 1999
%E Edited by _Dean Hickerson_, Sep 17 2002
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