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A056171 Pi(n)-pi(floor(n/2)), where pi is A000720. 14
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

Also, the number of unitary prime divisors of n!. A prime divisor of n is unitary iff its exponent is 1 in the prime power factorization of n. In general GCD(p, n/p) = 1 or p. Here we count the cases when GCD(p, n/p)=1.

A unitary prime divisor of n! is >= n/2, hence their number is pi(n)-pi(n/2). [Peter Luschny, Mar 13 2011].

See also 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

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

with(numtheory); [seq(pi(n)-pi(floor(n/2)), n=1..130)]; # from N. J. A. Sloane, Sep 01 2015

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}]

Table[PrimePi[n]-PrimePi[Floor[n/2]], {n, 100}] (* Harvey P. Dale, Sep 01 2015 *)

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

EXTENSIONS

Simplified definition. - N. J. A. Sloane, Sep 01 2015

STATUS

approved

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Last modified September 29 04:22 EDT 2016. Contains 276609 sequences.