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

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

Offset

1,1

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

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

Table of n, a(n) for n=1..22.

Paul Erdős, Problems and results on number theoretic properties of consecutive integers and related questions, Proc. 5th Manitoba Conf. Numerical Math., Congress. Num. 16 (1975), 25-44.

Laurent Habsieger, Explicit Bounds For The Diophantine Equation A!B! = C!, arXiv:1903.08370 [math.NT], 2019.

Example

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

Prog

(PARI) is_A109095(n) = my(m=1, f=n!); while(n-->m, while(n!<f, f\=m++); n!==f && return(m>2));
select(is_A109095, [0..777]) \\ M. F. Hasler, Jan 19 2023

Crossrefs

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

Sequence in context: A108899 A074289 A109099 * A349692 A351446 A323107

Adjacent sequences: A109092 A109093 A109094 * A109096 A109097 A109098

Keyword

nonn

Author

Jud McCranie, Jun 19 2005

Extensions

Definition corrected by Jon E. Schoenfield, Jul 02 2010

More terms from M. F. Hasler, Jan 19 2023

Status

approved