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A281498 Least k >= 0 such that 2^k * n! is not a refactorable number. 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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?

LINKS

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

EXAMPLE

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

PROG

(PARI) isA033950(n) = n % numdiv(n) == 0;

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

(PARI) 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

CROSSREFS

Cf. A033950, A281291.

Sequence in context: A156256 A029406 A158461 * A118269 A144152 A265674

Adjacent sequences:  A281495 A281496 A281497 * A281499 A281500 A281501

KEYWORD

nonn

AUTHOR

Altug Alkan, Jan 23 2017

STATUS

approved

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Last modified March 5 10:43 EST 2021. Contains 341822 sequences. (Running on oeis4.)