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

2, 4, 8, 16, 256, 65536

Offset

1,1

Comments

See Conjecture 47 and Theorem 51 in Zelinsky's paper for related points.

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)

Also note that Luca & Young paper gives a proof for n! is a refactorable number for all n > 5.

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.

Links

Table of n, a(n) for n=1..6.

Florian Luca and Paul Thomas Young, On the number of divisors of n! and of the Fibonacci numbers

S. Colton, Refactorable Numbers - A Machine Invention, J. Integer Sequences, Vol. 2, 1999.

Joshua Zelinsky, Tau Numbers: A Partial Proof of a Conjecture and Other Results, Journal of Integer Sequences, Vol. 5 (2002), Article 02.2.8.

Example

8 is a term since d(2*8!) = 2^2 * 3^3 does not divide 2 * 8! = 2^8 * 3^2 * 5 * 7.

Prog

(PARI) isA033950(n) = n % numdiv(n) == 0;
is(n) = !isA033950(2*n!);

Crossrefs

Cf. A019434, A033950, A052849, A281498.

Sequence in context: A321532 A061581 A046251 * A346644 A164312 A068806

Adjacent sequences: A281288 A281289 A281290 * A281292 A281293 A281294

Keyword

nonn,more

Author

Altug Alkan, Jan 23 2017

Status

approved