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A281291 Numbers n such that 2*n! is not a refactorable number.

%I A281291 #48 Jun 11 2020 23:57:01
%S A281291 2,4,8,16,256,65536
%N A281291 Numbers n such that 2*n! is not a refactorable number.
%C A281291 See Conjecture 47 and Theorem 51 in Zelinsky's paper for related points.
%C A281291 In Theorem 51 Zelinsky gives a technical result which almost implies that for all sufficiently large n, n! is a refactorable number. (Corrected by _Joshua Zelinsky_, May 15 2020)
%C A281291 Also note that Luca & Young paper gives a proof for n! is a refactorable number for all n > 5.
%C A281291 This sequence focuses on the 2 * n! and we cannot say that 2 * n! is refactorable for all sufficiently large n at the moment. This is because if 2^(2^k) + 1 is a Fermat prime (A019434), then 2^(2^k) is a term of this sequence and it is not known yet sequence of Fermat primes is finite or not.
%H A281291 Florian Luca and Paul Thomas Young, <a href="https://pdfs.semanticscholar.org/c341/206d68979e1668fc470f668868ccd3b69bcc.pdf">On the number of divisors of n! and of the Fibonacci numbers</a>
%H A281291 S. Colton, <a href="https://cs.uwaterloo.ca/journals/JIS/colton/joisol.html">Refactorable Numbers - A Machine Invention</a>, J. Integer Sequences, Vol. 2, 1999.
%H A281291 Joshua Zelinsky, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL5/Zelinsky/zelinsky9.html">Tau Numbers: A Partial Proof of a Conjecture and Other Results</a>, Journal of Integer Sequences, Vol. 5 (2002), Article 02.2.8.
%e A281291 8 is a term since d(2*8!) = 2^2 * 3^3 does not divide 2 * 8! = 2^8 * 3^2 * 5 * 7.
%o A281291 (PARI) isA033950(n) = n % numdiv(n) == 0;
%o A281291 is(n) = !isA033950(2*n!);
%Y A281291 Cf. A019434, A033950, A052849, A281498.
%K A281291 nonn,more
%O A281291 1,1
%A A281291 _Altug Alkan_, Jan 23 2017