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A056171 Number of unitary prime divisors of n!. 12
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 (list; graph; refs; listen; history; text; internal format)
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 non-unitary 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 * A076755 A106490 A122375

Adjacent sequences:  A056168 A056169 A056170 * A056172 A056173 A056174

KEYWORD

nonn,easy

AUTHOR

Labos Elemer, Jul 27 2000

STATUS

approved

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Last modified April 17 20:01 EDT 2014. Contains 240655 sequences.