A359636 -id:A359636 - cp's OEIS frontend

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

N. J. A. Sloane

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

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

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).

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

Cf. A000142, A034878, A093373 (complement), A344438 (characteristic function).
Union of A344181 and A344179. Subsequence of A025487 (see also A064783).
See also A359636 and A359751.

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

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

For[p=0; a=f=Table[n!, {n, 1, 8}], p=!=a, p=a; a=Select[Union@@Outer[Times, f, a], #<=8!&]]; a

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

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

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

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

Edited by Dean Hickerson, Sep 17 2002

A034878 Numbers k such that k! can be written as the product of smaller factorials.

Original entry on oeis.org

1, 4, 6, 8, 9, 10, 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

Offset: 1

Erich Friedman

Except for the numbers 2, 9 and 10 this sequence is conjectured to be the same as A001013.
Every r! is a member for r>2, for (r!)! = (r!)*(r!-1)!. - Amarnath Murthy, Sep 11 2002
By Murthy's trick, if k>2 is a product of factorials then k is a term. So half of the above conjecture is true: A001013 is a subsequence except for the number 2. - Jonathan Sondow, Nov 08 2004
If there exists another term of this sequence not also in A001013, it must be >= 100000. - Charlie Neder, Oct 07 2018
An additional term of this sequence not in A001013 must be > 5000000. Can it be shown that no such terms exist using results on consecutive smooth numbers? - Charlie Neder, Jan 14 2019

			1! = 0! (or, 1! is the empty product), 4! = 2!*2!*3!, 6! = 3!*5!, 8! = (2!)^3*7!, 9! = 2!*3!*3!*7!, 10! = 6!*7!, etc.

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

Charlie Neder, Table of n, a(n) for n = 1..222
Eric Weisstein's World of Mathematics, Factorial Products
Index entries for sequences related to factorial numbers

Cf. A000142, A075082, A001013.

A359637 a(n) is the least odd prime not in A001359 such that all subsequent composites in the gap up to the next prime have at least n prime factors, counted with multiplicity.

Original entry on oeis.org

7, 97, 349, 13309, 33613, 5594749, 84477247, 1524981247, 60924074749

Offset: 2

Hugo Pfoertner, Jan 16 2023

			a(2) = 7: 8 = 2^3, 9 = 3^2, 10 = 2*5 all have at least the minimum number of 2 prime factors;
a(3) = 97: 98 = 2*7^2, 99 = 3^2*11, 100 = 2^2*5^2 have a minimum of 3 prime factors;
a(4) = 349: 350 = 2*5^2*7, 351 = 3^3*13, 352 = 2^5*11 have a minimum of 4 prime factors.

Cf. A001222, A001359, A062502, A359636.

a359637(maxp) = {my (k=2, pp=3); forprime (p=5, maxp, my(mi=oo); if (p-pp>2, for (j=pp+1, p-1, my(mo=bigomega(j)); if (mo=k, print1(pp,", "); k++)); pp=p)};
a359637(10^8)

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

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A034878 Numbers k such that k! can be written as the product of smaller factorials.

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A359637 a(n) is the least odd prime not in A001359 such that all subsequent composites in the gap up to the next prime have at least n prime factors, counted with multiplicity.

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PARI

A359638 a(n) is the least odd prime not in A001359 such that all subsequent composites in the gap up to the next prime have exactly n prime factors, counted with multiplicity.

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PARI