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A056171
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Number of unitary prime divisors of n!.
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11
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0, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 3, 2, 2, 2, 3, 3, 4, 4, 4, 3, 4, 4, 4, 3, 3, 3, 4, 4, 5, 5, 5, 4, 4, 4, 5, 4, 4, 4, 5, 5, 6, 6, 6, 5, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 6, 7, 7, 8, 7, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 10, 9, 9, 9, 9, 9, 10, 10, 10, 9, 10, 10, 10, 9, 9, 9, 10, 10, 10, 10, 10, 9, 9, 9, 10, 10
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OFFSET
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1,3
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COMMENTS
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A unitary prime divisor for n! is not smaller than n/2, hence a(n)=PrimePi(n)-PrimePi(n/2) [Peter Luschny, Mar 13 2011].
See the references and links mentioned in A143227. [From Jonathan Sondow, Aug 03 2008]
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LINKS
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Daniel Forgues, Table of n, a(n) for n=1..100000
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FORMULA
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A prime divisor of n is unitary iff its exponent is 1 in prime power factorization of n. In general GCD(p, n/p)=1 or p. Cases are counted when GCD(p, n/p)=1.
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EXAMPLE
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10!=2.2.2.2.2.2.2.2.3.3.3.3.5.5.7 The only unitary prime divisor is 7, so a(10)=1, while 10! has 3 non-unitary prime divisors.
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MAPLE
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A056171 := n -> nops(select(isprime, [$iquo(n, 2)+1..n])):
seq(A056171(i), i=1..98); - Peter Luschny, Mar 13 2011
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MATHEMATICA
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s=0; Table[If[PrimeQ[k], s++]; If[PrimeQ[k/2], s--]; s, {k, 100}]
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CROSSREFS
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Cf. A001221, A034444, A000720, A048105, A048656, A048657.
Cf. A014085, A060715, A104272, A143223, A143224, A143225, A143226, A143227. [From Jonathan Sondow, Aug 03 2008]
Sequence in context: A163377 A163109 A128428 * A076755 A106490 A122375
Adjacent sequences: A056168 A056169 A056170 * A056172 A056173 A056174
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KEYWORD
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nonn,easy
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AUTHOR
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Labos E. (labos(AT)ana.sote.hu), Jul 27 2000
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STATUS
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approved
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