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A001013 Jordan-Polya numbers: products of factorial numbers A000142.
(Formerly M0993 N0372)
27
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)*(6x^2-6x+(6x-3)a(n)+2a(n)^2+1) = N^2 has an integer solution. - Ralf Stephan, Dec 04 2004

REFERENCES

R. K. Guy, Unsolved Problems in Number Theory, B23.

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 R. Zumkeller, Table of n, a(n) for n = 1..10000, first 987 terms by T. D. Noe

Robert A. Melter, Autometrized unary algebras, J. Combinatorial Theory 5 1968 21-29.

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)

# For the function 'prod_hull' see 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

CROSSREFS

Cf. A034878, A093373 (complement).

Sequence in context: A250270 A062847 * A115746 A025610 A242101 A131117

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 July 3 19:55 EDT 2015. Contains 259178 sequences.