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%I #19 Jan 25 2017 12:13:05
%S 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,
%T 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,
%U 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
%N Least k >= 0 such that 2^k * n! is not a refactorable number.
%C 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.
%C It is known that a(n) > 0 for all n > 5. See related comment in A281291.
%C 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, ...
%C What is the asymptotic behavior of this sequence?
%H Charles R Greathouse IV, <a href="/A281498/b281498.txt">Table of n, a(n) for n = 1..10000</a>
%H 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 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 a(2^8) = 1 because 2 * (2^8)! is not a refactorable number.
%o (PARI) isA033950(n) = n % numdiv(n) == 0;
%o a(n) = my(k=0); while (isA033950 (2^k*n!), k++); k;
%o (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
%Y Cf. A033950, A281291.
%K nonn
%O 1,1
%A _Altug Alkan_, Jan 23 2017
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