A281291 -id:A281291 - cp's OEIS frontend

A281498 Least k >= 0 such that 2^k * n! is not a refactorable number.

2, 1, 0, 1, 0, 2, 4, 1, 2, 2, 4, 2, 6, 5, 5, 1, 3, 2, 6, 4, 4, 3, 9, 6, 6, 5, 5, 3, 5, 4, 10, 5, 5, 4, 4, 2, 6, 5, 5, 2, 4, 3, 7, 5, 5, 4, 10, 6, 6, 5, 5, 3, 9, 8, 8, 5, 5, 4, 6, 4, 10, 9, 9, 3, 3, 2, 6, 4, 4, 3, 5, 2, 8, 7, 7, 5, 5, 4, 8, 4, 3, 3, 9, 7, 7, 6, 6, 3, 11, 10, 10, 8, 8, 7, 7, 2, 6, 5, 5, 3

Offset: 1

Altug Alkan, Jan 23 2017

nonn

Such k always exists. There are infinitely many values of k such that A011371(n) + k + 1 does not divide 2^k * n!; i.e., prime q = A011371(n) + k + 1 > n.
It is known that a(n) > 0 for all n > 5. See related comment in A281291.
The values of a(A000040(n)) are 1, 0, 0, 4, 4, 6, 3, 6, 9, 5, 10, 6, 4, 7, 10, 9, 6, 10, 6, 5, 8, 8, 9, ...
What is the asymptotic behavior of this sequence?

			a(2^8) = 1 because 2 * (2^8)! is not a refactorable number.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
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

Cf. A033950, A281291.

isA033950(n) = n % numdiv(n) == 0;
a(n) = my(k=0); while (isA033950 (2^k*n!), k++); k;

a(n)=my(N=n!,o=valuation(N,2),d=numdiv(N>>=o),k); while((N<<(o+k))%(d*(o+k+1))==0, k++); k \\ Charles R Greathouse IV, Jan 25 2017

cp's OEIS Frontend

A281498 Least k >= 0 such that 2^k * n! is not a refactorable number.

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