cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 18 results. Next

A075082 Numbers n such that n! is a product of distinct factorials k!*l!*m!*... with k, l, m, etc. < n.

Original entry on oeis.org

1, 6, 10, 12, 16, 24, 48, 120, 144, 240, 288, 720, 1440, 2880, 4320, 5040, 5760, 8640, 10080, 17280, 30240, 34560, 40320, 60480, 80640, 86400, 103680, 120960, 172800, 207360, 241920, 362880, 483840, 518400, 604800, 725760, 967680, 1036800
Offset: 1

Views

Author

Amarnath Murthy, Sep 11 2002

Keywords

Comments

r! is a member for r>2, since (r!)! = (r!)*(r!-1)!.
Subsequence of A034878 (all n such that n! is a product of smaller factorials). It is conjectured that A034878 and A001013 (Jordan-Polya numbers = products of factorials) are the same sequence (except for the numbers 2, 9 and 10). If this is true, then obviously A075082 (without the number 10) is also a subsequence of A001013. On the other hand, this special case of the conjecture might be easier to prove. (a(n)!)^2 is a member of A058295 (products of distinct factorials); for example, (6!)^2 = 6!*5!*3!. - Jonathan Sondow, Dec 21 2004
May be the same as A058295 except for 2, 10 and 16. - Jud McCranie, Jun 13 2005
By using similar logic, r!s!t! is a member for at least two, all distinct r,s,t,... > 1. - Robert G. Wilson v, Jan 27 2006
Except for 1, 10 & 16, all the members are of the form immediately above. - Robert G. Wilson v, Jan 27 2006
Except for 10 and 16, all members, n, have as the greatest factorial in is product representation of n, n-1. - Robert G. Wilson v, Jan 27 2006
Theorem, for n to be a member of A075082, then the largest distinct factorial, m!, less than n! must not be less than the greatest prime less than n. - Robert G. Wilson v, Jan 27 2006

Examples

			1! = 0!, 6! = 5!*3!, 10! = 7!*6!, 12! = 11!*3!*2!, 16! = 14!*5!*2!,
24! = 23!*4!, 48! = 47!*4!*2!, 120! = 119!*5!, 144! = 143! *4!*3!,
240! = 239!*5!*2!, 288! = 287!*4!*3!*2!, 720! = 719!*6!,
1440! = 1439!6!*2!, etc.

References

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

Links

Crossrefs

Cf. A034878, A001013, A058295.

Programs

  • Mathematica
    (* first do *) Needs["DiscreteMath`Combinatorica`"] (* then *) s = Sort[ Table[ Times @@ Factorial /@ UnrankSubset[n, Table [i, {i, 2, 12}]], {n, 2047}]]; f[n_] := Block[{k = Prime[ PrimePi [n]]}, While[k < n && Position[s, Product[i, {i, k + 1, n}]] == {}, k+ + ]; If[k == n, 0, k]]; Do[a = f[n]; If[a != 0, Print[{n, a}]], {n, 3, 1210000}] (* Robert G. Wilson v, Jun 20 2005 *)

Extensions

Corrected and extended by Jud McCranie, Sep 13 2002
More terms from Jud McCranie, Jun 13 2005
a(25)-a(39) proposed by Robert G. Wilson v, Jun 18 2005, confirmed by David Wasserman, Dec 30 2005

A003135 n! is a nontrivial product of factorials. It is conjectured that the list is complete.

Original entry on oeis.org

9, 10, 16
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

A "nontrivial" solution is one in which the largest x! in the product of a(n)! is such that x < a(n)-1. There are no other terms < 10^5. - Jud McCranie, Jun 15 2005

Examples

			9! = 2! * 3! * 3! * 7! and 7 < 9-1, so 9 is in the sequence.
10! = 6! * 7! or 10! = 3! * 5! * 7! and 7 < 10-1, so 10 is in the sequence.
16! = 2! * 5! * 14! and 14 < 16-1, so 16 is in the sequence.

References

  • R. K. Guy, "Unsolved Problems in Number Theory", section B23.

Links

Crossrefs

Cf. A034878, A001013, A058295, A075082, A109095, A109096, A109097, A109098.

A109095 Numbers N such that N! is the product of exactly two smaller factorials (larger than 1).

Original entry on oeis.org

6, 10, 24, 120, 720, 5040, 40320, 362880, 3628800, 39916800, 479001600, 6227020800, 87178291200, 1307674368000, 20922789888000, 355687428096000, 6402373705728000, 121645100408832000, 2432902008176640000, 51090942171709440000, 1124000727777607680000, 25852016738884976640000
Offset: 1

Views

Author

Jud McCranie, Jun 19 2005

Keywords

Comments

N = x! is considered to be a trivial solution because then N! = N*(N-1)! = x!*(N-1)!. Therefore every factorial appears in this sequence.
All terms except a(2) = 10 appear to be trivial solutions. (From Erdős's paper this is known as Surányi's conjecture.)
Habsieger established that the least nontrivial solution must have N > 10^3000. - M. F. Hasler, Jan 19 2023

Examples

			10! = 6! * 7!, so 10 is in the sequence.

References

  • Richard K. Guy, Unsolved Problems in Number Theory, B23 Equal products of factorials, Springer, Third Edition, 2004, p. 123.
  • Laurent Habsieger, Explicit bounds for the Diophantine equation A!B! = C!, Fibonacci Quarterly (2019), 57, 1.

Links

Crossrefs

Cf. A000142, A034878, A001013, A003135, A058295, A075082, A109096, A109097.

Programs

Extensions

Definition corrected by Jon E. Schoenfield, Jul 02 2010
More terms from M. F. Hasler, Jan 19 2023

A109096 Numbers n such that n! is the product of exactly three smaller factorials.

Original entry on oeis.org

4, 10, 12, 16, 36, 48, 144, 240, 576, 720, 1440, 2880, 4320, 10080, 14400, 17280, 30240, 80640, 86400
Offset: 1

Views

Author

Jud McCranie, Jun 19 2005

Keywords

Examples

			144! = 3! * 4! * 143!, so 144 is in the sequence.

Crossrefs

Cf. A034878, A001013, A003135, A058295, A075082, A109095, A109097.

Extensions

Definition corrected by Jon E. Schoenfield, Jul 02 2010

A109097 Numbers n such that n! is the product of exactly four smaller factorials.

Original entry on oeis.org

8, 9, 24, 72, 96, 216, 288, 480, 864, 1152, 1440, 2880, 3456, 4320, 5760, 8640, 13824, 17280, 20160, 25920, 28800, 34560, 60480, 69120, 86400
Offset: 1

Views

Author

Jud McCranie, Jun 19 2005

Keywords

Examples

			86400! = 3! * 5! * 5! * 86399!, so 86400 is in the sequence.

Crossrefs

Cf. A034878, A001013, A003135, A058295, A075082, A109095, A109096.

Extensions

Definition corrected by Jon E. Schoenfield, Jul 02 2010

A109098 Numbers n such that n! is the product of exactly 5 smaller factorials (greater than 1).

Original entry on oeis.org

16, 48, 144, 192, 432, 576, 960, 1296, 1728, 2304, 2880, 5184, 5760, 6912, 8640, 11520, 17280, 20736, 25920, 27648, 34560, 40320, 51840, 57600, 69120, 82944
Offset: 1

Views

Author

Jud McCranie, Jun 19 2005

Keywords

Examples

			144! = 2! * 2! * 3! * 3! * 143!, so 144 is in the sequence.

Crossrefs

Cf. A034878, A001013, A003135, A058295, A075082, A109095, A109096, A109097.

A109100 Numbers n such that n! is the product of exactly 2 smaller factorials (greater than 1).

Original entry on oeis.org

10, 16, 24, 48, 96, 144, 192, 288, 384, 720, 768, 864, 1440, 1536, 1728, 3072, 4320, 5184, 6144, 8640, 10368, 12288, 24576, 25920, 31104, 40320, 49152, 51840, 62208, 80640, 86400, 98304
Offset: 1

Views

Author

Jud McCranie, Jun 19 2005

Keywords

Examples

			10! = 3! * 5! * 7! = 6! * 7!, so 10 is in the sequence.

Crossrefs

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

A109101 Numbers n such that n! is the product of exactly 3 smaller factorials (greater than 1).

Original entry on oeis.org

576, 1152, 2304, 2880, 3456, 4608, 5760, 6912, 9216, 11520, 18432, 20736, 23040, 36864, 41472, 46080, 73728, 92160
Offset: 1

Views

Author

Jud McCranie, Jun 19 2005

Keywords

Crossrefs

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

A109102 Numbers n such that n! is the product of exactly 4 smaller factorials (greater than 1).

Original entry on oeis.org

13824, 17280, 27648, 34560, 55296, 82944
Offset: 1

Views

Author

Jud McCranie, Jun 19 2005

Keywords

Crossrefs

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

A109099 Numbers n such that n! can be expressed as the product of smaller factorials > 2.

Original entry on oeis.org

6, 10, 24, 36, 120, 144, 216, 576, 720, 864, 1296, 2880, 3456, 4320, 5040, 5184, 7776, 13824, 14400, 17280, 20736, 25920, 30240, 31104, 40320, 46656, 69120, 82944, 86400
Offset: 1

Views

Author

Jud McCranie, Jun 19 2005

Keywords

Examples

			86400! = 5! * 6! * 86399!, so 86400 is in the sequence.

Crossrefs

Cf. A034878, A001013, A003135, A058295, A075082, A109095, A109096, A109097, A109098.

Extensions

Definition corrected by Jon E. Schoenfield, Jul 02 2010
Showing 1-10 of 18 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.