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

Author

Jud McCranie, Jun 19 2005

Keywords

• nonn

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

• 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.

Crossrefs

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

Programs

PARI

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

Extensions

Definition corrected by Jon E. Schoenfield, Jul 02 2010

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