A001013 Jordan-Polya numbers: products of factorial numbers A000142.

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

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• easy

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

Examples

			864 = (3!)^2*4!.
		

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.
• George Pólya, Kombinatorische Anzahlbestimmungen für Gruppen, Graphen und chemische Verbindungen, Acta Mathematica, Vol. 68 (1937), pp. 145-254; alternative link.
• Eric Weisstein's World of Mathematics, Factorial Products.
• Index entries for sequences related to factorial numbers
• Index entries for sequences related to trees

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.

Programs

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

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

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

Sage

# uses[prod_hull from A246663]
prod_hull(factorial, 5760) # Peter Luschny, Sep 09 2014

Extensions

More terms, formula from Christian G. Bower, Dec 15 1999

Edited by Dean Hickerson, Sep 17 2002