A058295 -id:A058295 - cp's OEIS frontend

A109103 Smallest a(n) such that a(n)! can be expressed as the product of smaller factorials, using n distinct factorials greater than 1 (with repetitions allowed).

4, 9, 288, 34560

Offset: 2

Jud McCranie, Jun 19 2005

nonn

			34560! = 2! * 3! * 4! * 5! * 34559!, using five different factorials, so a(5)=34560.

Cf. A034878, A001013, A003135, A058295, A075082, A109095, A109096, A109097, A109098, A109099, A109100, A109101, A109102.

A101977 Number of products of distinct factorials not exceeding n!.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 15, 22, 31, 43, 58, 74, 97, 131, 171, 222, 277, 349, 447, 564, 698, 868, 1074, 1321, 1601, 1967, 2398, 2911, 3513, 4235, 5083, 6071, 7242, 8637, 10229, 12102, 14293, 16848, 19802, 23271, 27276, 31846, 37132, 43196, 50191, 58238, 67425, 77946

Offset: 1

Jonathan Sondow, Dec 22 2004

nonn

a(n) is the position of n! in A058295 (products of distinct factorials). a(n) < A101976(n) for n > 2 and a(n) > A101978(n) for n > 10.

			a(4) = 5 because 5 products of distinct factorials do not exceed 4!, namely, 1, 2, 6, 12 and 24.

Index entries for sequences related to factorial numbers.

Cf. A000142, A058295, A101976, A101978.

d[k_] := (m=1; With[{p=With[{s=Subsets[Table[n!, {n, k}]]}, Sort[Table[Apply[Times, s[[n]]], {n, Length[s]}]]]}, While[p[[m]]<=k!, m++ ]; Length[Union[Take[p, m-1]]]]);Table[d[k], {k, 19}]

a(20)-a(48) from Donovan Johnson, May 30 2012

A216152 A205957(n) where n is a nonprime number.

Original entry on oeis.org

1, 2, 12, 48, 144, 1440, 34560, 483840, 7257600, 58060800, 3135283200, 125411328000, 2633637888000, 57940033536000, 5562243219456000, 27811216097280000, 723091618529280000, 6507824566763520000, 364438175738757120000, 327994358164881408000000

Offset: 1

Peter Luschny, Sep 02 2012

nonn

The distinct values of A205957. Partial products of A216153.

a(1),...,a(10) are highly totient numbers (A097942) and products of distinct factorials (A058295). The author conjectures that this is true in general.

Peter Luschny, The von Mangoldt Transformation.

Cf. A051451.

A205957[n_] := Exp[-Sum[MoebiusMu[p] Log[k/p], {k, 1, n}, {p, FactorInteger[k][[All, 1]]}]];
Table[A205957[n], {n, 0, 30}] // DeleteDuplicates (* Jean-François Alcover, Jul 08 2019 *)

# sorted(list(set([A205957(n) for n in (0..31)])))
def A216152_list(n) :
    C = filter(lambda k: not is_prime(k), (1..n))
    return [A205957(c) for c in C]
A216152_list(31)

a(n) = A205957(A018252(n)).

A334174 Numbers that can be written as a product of two or more consecutive factorial numbers.

Original entry on oeis.org

1, 2, 12, 144, 288, 2880, 17280, 34560, 86400, 2073600, 3628800, 12441600, 24883200, 203212800, 435456000, 10450944000, 14631321600, 62705664000, 125411328000, 146313216000, 1316818944000, 17557585920000, 73741860864000, 144850083840000, 421382062080000

Offset: 1

Ilya Gutkovskiy, Apr 17 2020

nonn

			    1 = 0! * 1!;
    2 = 1! * 2!;
   12 = 2! * 3!;
  144 = 3! * 4!;
  288 = 2! * 3! * 4!.

David A. Corneth, Table of n, a(n) for n = 1..1803 (terms < 10^1000)
Eric Weisstein's World of Mathematics, Factorial Products
Index entries for sequences related to factorial numbers

Cf. A000142, A000178, A001013, A010790, A034882, A045619, A058295, A334175.

A109104 Numbers n such that n! can be expressed as the product of the factorials of prime numbers, repetitions allowed.

Original entry on oeis.org

4, 6, 8, 9, 10, 12, 24, 32, 48, 72, 128, 192, 240, 384, 432, 480, 720, 864, 1152, 1440, 2592, 2880, 5040, 6144, 6912, 8192, 10080, 11520, 15360, 15552, 23040, 25920, 27648, 51840, 62208, 69120, 73728, 86400

Offset: 1

Jud McCranie, Jun 19 2005

nonn

			10! = 3! * 5! * 7!, so 10 is in the sequence.

Cf. A034878, A001013, A003135, A058295, A075082, A109095, A109096, A109097, A109098, A109099, A109100, A109101, A109102, A109103.

A255937 Number of distinct products of distinct factorials up to n!.

Original entry on oeis.org

1, 1, 2, 4, 8, 16, 28, 56, 108, 204, 332, 664, 1114, 2228, 4078, 7018, 11402, 22804, 40638, 81276, 140490, 230328, 391544, 783088, 1287034, 2273676, 3903626, 6837760, 10368184, 20736368, 34081198, 68162396

Offset: 0

Charles R Greathouse IV, Mar 11 2015

			a(3) = |{1!, 2!, 3!, 2!*3!}| = |{1, 2, 6, 12}| = 4.

Paul Erdős and Ron L. Graham, On products of factorials, Bull. Inst. Math. Acad. Sinica 4:2 (1976), pp. 337-355.

Cf. A058295, A000142, A001013, A060957.

s:= proc(n) option remember; (f-> `if`(n=0, {f},
      map(x-> [x, x*f][], s(n-1))))(n!)
    end:
a:= n-> nops(s(n)):
seq(a(n), n=0..20);  # Alois P. Heinz, Mar 16 2015

a[n_] := a[n] = If[n == 0, 1, If[PrimeQ[n], 2 a[n-1], Times @@@ ((Subsets[Range[n]] // Rest) /. k_Integer -> k!) // Union // Length]];
Table[Print[n, " ", a[n]]; a[n], {n, 0, 23}] (* Jean-François Alcover, May 01 2022 *)

a(n)=my(v=[1],N=n!); for(k=2,n-1, v=Set(concat(v,v*k!))); #v + sum(i=1,#v, !setsearch(v,N*v[i]))

Erdős and Graham prove that log a(n) ~ n log log n/log n.

a(p) = 2*a(p-1) for prime p. - Jon E. Schoenfield, Apr 01 2015

More terms from Alois P. Heinz, Mar 16 2015

a(31) (=2*a(30)) from Jon E. Schoenfield, Apr 01 2015

A309841 If n = Sum (2^e_k) then a(n) = Product ((e_k + 2)!).

Original entry on oeis.org

1, 2, 6, 12, 24, 48, 144, 288, 120, 240, 720, 1440, 2880, 5760, 17280, 34560, 720, 1440, 4320, 8640, 17280, 34560, 103680, 207360, 86400, 172800, 518400, 1036800, 2073600, 4147200, 12441600, 24883200, 5040, 10080, 30240, 60480, 120960, 241920, 725760, 1451520, 604800

Offset: 0

Ilya Gutkovskiy, Aug 19 2019

			21 = 2^0 + 2^2 + 2^4 so a(21) = 2! * 4! * 6! = 34560.

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A000142, A000178, A019565, A029930, A058295, A059590, A121663, A283477.

a:= n-> (l-> mul((i+1)!^l[i], i=1..nops(l)))(convert(n, base, 2)):
seq(a(n), n=0..40);  # Alois P. Heinz, Feb 10 2020

nmax = 40; CoefficientList[Series[Product[(1 + (k + 2)! x^(2^k)), {k, 0, Floor[Log[2, nmax]] + 1}], {x, 0, nmax}], x]
a[0] = 1; a[n_] := (Floor[Log[2, n]] + 2)! a[n - 2^Floor[Log[2, n]]]; Table[a[n], {n, 0, 40}]

a(n)={vecprod([(k+1)! | k<-Vec(select(b->b, Vecrev(digits(n, 2)), 1))])} \\ Andrew Howroyd, Aug 19 2019

G.f.: Product_{k>=0} (1 + (k + 2)! * x^(2^k)).
a(0) = 1; a(n) = (floor(log_2(n)) + 2)! * a(n - 2^floor(log_2(n))).
a(2^(k-1)-1) = A000178(k).

A356639 Number of integer sequences b with b(1) = 1, b(m) > 0 and b(m+1) - b(m) > 0, of length n which transform under the map S into a nonnegative integer sequence. The transform c = S(b) is defined by c(m) = Product_{k=1..m} b(k) / Product_{k=2..m} (b(k) - b(k-1)).

Original entry on oeis.org

1, 1, 3, 17, 155, 2677, 73327, 3578339, 329652351

Offset: 1

Thomas Scheuerle, Aug 19 2022

This sequence can be calculated by a recursive algorithm:
Let B1 be an array of finite length, the "1" denotes that it is the first generation. Let B1' be the reversed version of B1. Let C be the element-wise product C = B1 * B1'. Then B2 is a concatenation of taking each element of B1 and add all divisors of the corresponding element in C. If we start with B1 = {1} then we get this sequence of arrays: B2 = {2}, B3 = {3, 4, 6}, ... . a(n) is the length of the array Bn. In short the length of Bn+1 and so a(n+1) is the sum over A000005(Bn * Bn').
The transform used in the definition of this sequence is its own inverse, so if c = S(b) then b = S(c). The eigensequence is 2^n = S(2^n).
There exist some transformation pairs of infinite sequences in the database:
  A000027 <--> A000142; A000079 <--> A000079; A000045 <--> A001654;
  A026549 <--> A038754; A100071 <--> A001405; A058295 <--> A------;
  A111286 <--> A098011; A093968 <--> A205825; A166447 <--> A------;
  A098558 <--> A004277; A010551 <--> A087811; A100538 <--> A329227;
  A079352 <--> A------; A082458 <--> A------; A008233 <--> A264635;
  A138278 <--> A------; A006501 <--> A264557; A336496 <--> A------;
  A019464 <--> A------; A062112 <--> A------; A171647 <--> A359039;
  A279312 <--> A------; A031923 <--> A------.
These transformation pairs are conjectured:
  A137326 <--> A------; A066332 <--> A300902; A208147 <--> A308546;
  A057895 <--> A------; A349080 <--> A------; A019442 <--> A------;
  A349079 <--> A------.
("A------" means not yet in the database.)
Some sequences in the lists above may need offset adjustment to force a beginning with 1,2,... in the transformation.
If we allowed signed rational numbers, further interesting transformation pairs could be observed. For example, 1/n will transform into factorials with alternating sign. 2^(-n) transforms into ones with alternating sign and 1/A000045(n) into A000045 with alternating sign.

			a(4) = 17. The 17 transformation pairs of length 4 are:
  {1, 2, 3, 4}  = S({1, 2, 6, 24}).
  {1, 2, 3, 5}  = S({1, 2, 6, 15}).
  {1, 2, 3, 6}  = S({1, 2, 6, 12}).
  {1, 2, 3, 9}  = S({1, 2, 6, 9}).
  {1, 2, 3, 12} = S({1, 2, 6, 8}).
  {1, 2, 3, 21} = S({1, 2, 6, 7}).
  {1, 2, 4, 5}  = S({1, 2, 4, 20}).
  {1, 2, 4, 6}  = S({1, 2, 4, 12}).
  {1, 2, 4, 8}  = S({1, 2, 4, 8}).
  {1, 2, 4, 12} = S({1, 2, 4, 6}).
  {1, 2, 4, 20} = S({1, 2, 4, 5}).
  {1, 2, 6, 7}  = S({1, 2, 3, 21}).
  {1, 2, 6, 8}  = S({1, 2, 3, 12}).
  {1, 2, 6, 9}  = S({1, 2, 3, 9}).
  {1, 2, 6, 12} = S({1, 2, 3, 6}).
  {1, 2, 6, 15} = S({1, 2, 3, 5}).
  {1, 2, 6, 24} = S({1, 2, 3, 4}).
b(1) = 1 by definition, b(2) = 1+1 as 1 has only 1 as divisor.
a(3) = A000005(b(2)*b(2)) = 3.
The divisors of b(2) are 1,2,4. So b(3) can be b(2)+1, b(2)+2 and b(2)+4.
a(4) = A000005((b(2)+1)*(b(2)+4)) + A000005((b(2)+2)*(b(2)+2)) + A000005((b(2)+4)*(b(2)+1)) = 17.

Cf. A000005.
Cf. A000027, A000045, A000079, A000142, A001405.
Cf. A001654, A004277, A006501, A008233, A019442.
Cf. A010551, A019464, A026549, A031923, A038754.
Cf. A057895, A058295, A062112, A066332, A100071.
Cf. A079352, A082458, A087811, A093968, A098011.
Cf. A098558, A100538, A111286, A166447, A137326.
Cf. A138278, A171647, A205825, A208147, A264557.
Cf. A264635, A308546, A329227, A336496, A349079.
Cf. A349080, A359039.

cp's OEIS Frontend

A109103 Smallest a(n) such that a(n)! can be expressed as the product of smaller factorials, using n distinct factorials greater than 1 (with repetitions allowed).

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A101977 Number of products of distinct factorials not exceeding n!.

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Mathematica

Extensions

A216152 A205957(n) where n is a nonprime number.

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Mathematica

Sage

Formula

A334174 Numbers that can be written as a product of two or more consecutive factorial numbers.

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A109104 Numbers n such that n! can be expressed as the product of the factorials of prime numbers, repetitions allowed.

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A255937 Number of distinct products of distinct factorials up to n!.

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Maple

Mathematica

PARI

Formula

Extensions

A309841 If n = Sum (2^e_k) then a(n) = Product ((e_k + 2)!).

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Maple

Mathematica

PARI

Formula

A356639 Number of integer sequences b with b(1) = 1, b(m) > 0 and b(m+1) - b(m) > 0, of length n which transform under the map S into a nonnegative integer sequence. The transform c = S(b) is defined by c(m) = Product_{k=1..m} b(k) / Product_{k=2..m} (b(k) - b(k-1)).

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