cp's OEIS Frontend

A308168 Numbers m that cannot be represented as a k-tuple factorial b!k for any b and k < m-1.

%I A308168 #40 Feb 16 2025 08:33:55
%S A308168 3,4,5,7,9,11,13,17,19,23,25,29,31,35,37,41,43,47,49,53,59,61,67,71,
%T A308168 73,77,79,83,89,97,101,103,107,109,113,121,127,131,137,139,143,149,
%U A308168 151,157,163,167,169,173,179,181,187,191,193,197,199,211,221,223,227,229,233,239,241,247,251,257,263
%N A308168 Numbers m that cannot be represented as a k-tuple factorial b!k for any b and k < m-1.
%C A308168 If k >= m-1, then every number can be represented as a multifactorial: m = m!k.
%C A308168 The sequence contains only primes and numbers of the form p*q, where p and q are both prime and satisfy the inequalities p >= q and p-q < q-1.
%C A308168 Proof: If m has exactly two prime factors p and q (p > q), but p and q do not satisfy the second inequality, then m = p!(p-q). If, on the other hand, m has at least three factors a, b and c, (a >= b >= c > 1, m = a*b*c), then a*b-c > c-1, so m = (a*b)!(a*b-c).
%C A308168 Moreover, the sequence contains all numbers of that form. Proof: If they could be represented as a multifactorial, then it would be a (p-q)-tuple factorial. But as the second inequality is true, q-(p-q) is positive, therefore q-(p-q) should also divide m. But m has only two prime factors p and q, so the assumption is wrong and sequence indeed contains all numbers of that form.
%C A308168 1 and 2 are not in the sequence, because (-1)- and 0-tuple factorials are not defined.
%C A308168 Squarefree semiprimes that are in this sequence (35, 77, 143, 187, 209, 221, ...) are all in A259282 and they are the only semiprimes there. (See the Echi and Ghanmi reference for a proof.) - _Elijah Beregovsky_, Feb 05 2020
%H A308168 O. Echi, N. Ghanmi, <a href="https://www.researchgate.net/publication/267677588_The_Korselt_set_of_pq">The Korselt set of pq</a>, International Journal of Number Theory, Vol. 8 (2012), 2, 299-309.
%H A308168 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Multifactorial.html">Multifactorial</a>
%e A308168 15 is not in the sequence because 15 = 1*3*5 = 5!!.
%e A308168 35 is in the sequence because 35 = 7*5 and 7-5 < 5-1.
%Y A308168 Cf. A129116, A259282.
%K A308168 nonn,easy
%O A308168 1,1
%A A308168 _Elijah Beregovsky_, May 15 2019