

A056171


Number of unitary prime divisors of n!.


13



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

1,3


COMMENTS

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]


LINKS

Daniel Forgues, Table of n, a(n) for n=1..100000


FORMULA

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.


EXAMPLE

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 nonunitary prime divisors.


MAPLE

A056171 := n > nops(select(isprime, [$iquo(n, 2)+1..n])):
seq(A056171(i), i=1..98);  Peter Luschny, Mar 13 2011


MATHEMATICA

s=0; Table[If[PrimeQ[k], s++]; If[PrimeQ[k/2], s]; s, {k, 100}]


CROSSREFS

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 * A238949 A076755 A106490
Adjacent sequences: A056168 A056169 A056170 * A056172 A056173 A056174


KEYWORD

nonn,easy


AUTHOR

Labos Elemer, Jul 27 2000


STATUS

approved



