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A001013 Jordan-Polya numbers: products of factorial numbers A000142.
(Formerly M0993 N0372)
52
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

T. D. Noe and 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

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).

Sequence in context: A096850 A250270 A062847 * A115746 A025610 A344181

Adjacent sequences:  A001010 A001011 A001012 * A001014 A001015 A001016

KEYWORD

nonn,nice,easy

AUTHOR

N. J. A. Sloane

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 January 21 00:46 EST 2022. Contains 350473 sequences. (Running on oeis4.)