cp's OEIS Frontend

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

%I A109095 #32 Jan 20 2023 09:03:52
%S A109095 6,10,24,120,720,5040,40320,362880,3628800,39916800,479001600,
%T A109095 6227020800,87178291200,1307674368000,20922789888000,355687428096000,
%U A109095 6402373705728000,121645100408832000,2432902008176640000,51090942171709440000,1124000727777607680000,25852016738884976640000
%N A109095 Numbers N such that N! is the product of exactly two smaller factorials (larger than 1).
%C A109095 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.
%C A109095 All terms except a(2) = 10 appear to be trivial solutions. (From Erdős's paper this is known as Surányi's conjecture.)
%C A109095 Habsieger established that the least nontrivial solution must have N > 10^3000. - _M. F. Hasler_, Jan 19 2023
%D A109095 Richard K. Guy, Unsolved Problems in Number Theory, B23 Equal products of factorials, Springer, Third Edition, 2004, p. 123.
%D A109095 Laurent Habsieger, Explicit bounds for the Diophantine equation A!B! = C!, Fibonacci Quarterly (2019), 57, 1.
%H A109095 Paul Erdős, <a href="http://www.renyi.hu/~p_erdos/1976-39.pdf">Problems and results on number theoretic properties of consecutive integers and related questions</a>, Proc. 5th Manitoba Conf. Numerical Math., Congress. Num. 16 (1975), 25-44.
%H A109095 Laurent Habsieger, <a href="https://arxiv.org/abs/1903.08370">Explicit Bounds For The Diophantine Equation A!B! = C!</a>, arXiv:1903.08370 [math.NT], 2019.
%e A109095 10! = 6! * 7!, so 10 is in the sequence.
%o A109095 (PARI) is_A109095(n) = my(m=1, f=n!); while(n-->m, while(n!<f,f\=m++); n!==f && return(m>2));
%o A109095 select(is_A109095, [0..777]) \\ _M. F. Hasler_, Jan 19 2023
%Y A109095 Cf. A000142, A034878, A001013, A003135, A058295, A075082, A109096, A109097.
%K A109095 nonn
%O A109095 1,1
%A A109095 _Jud McCranie_, Jun 19 2005
%E A109095 Definition corrected by _Jon E. Schoenfield_, Jul 02 2010
%E A109095 More terms from _M. F. Hasler_, Jan 19 2023