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A182910
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Number of unitary prime divisors of the swinging factorial (A056040) n$ = n! / floor(n/2)!^2.
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1
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0, 0, 1, 2, 2, 3, 1, 2, 3, 3, 1, 2, 3, 4, 3, 3, 4, 5, 4, 5, 4, 6, 5, 6, 5, 5, 4, 4, 3, 4, 5, 6, 7, 8, 6, 6, 7, 8, 7, 7, 8, 9, 9, 10, 9, 7, 6, 7, 7, 7, 7, 8, 7, 8, 8, 10, 11, 13, 12, 13, 11, 12, 11, 10, 11, 13, 12, 13
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OFFSET
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0,4
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COMMENTS
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A prime divisor of n is unitary iff its exponent is 1 in the prime power factorization of n. A unitary prime divisor of the swinging factorial n$ can be smaller than n/2. For n >= 30 the swinging factorial has more unitary prime divisors than the factorial and it has never less unitary prime divisors. Thus a(n) >= PrimePi(n) - PrimePi(n/2).
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LINKS
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Table of n, a(n) for n=0..67.
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EXAMPLE
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16$ = 2.3.3.5.11.13. So 16$ has one non-unitary prime divisor and a(16) = 4.
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MAPLE
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UnitaryPrimeDivisor := proc(f, n) local k, F; F := f(n):
add(`if`(igcd(iquo(F, k), k)=1, 1, 0), k=numtheory[factorset](F)) end;
A056040 := n -> n!/iquo(n, 2)!^2;
A182910 := n -> UnitaryPrimeDivisor(A056040, n);
seq(A182910(i), i=1..LEN);
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CROSSREFS
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Cf. A056171.
Sequence in context: A054707 A166269 A181648 * A055460 A067514 A115323
Adjacent sequences: A182907 A182908 A182909 * A182911 A182912 A182913
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KEYWORD
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nonn
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AUTHOR
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Peter Luschny, Mar 14 2011
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STATUS
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approved
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