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A281498
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Least k >= 0 such that 2^k * n! is not a refactorable number.
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2
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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
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OFFSET
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1,1
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COMMENTS
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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?
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LINKS
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EXAMPLE
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a(2^8) = 1 because 2 * (2^8)! is not a refactorable number.
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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