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A056171 a(n) = pi(n) - pi(floor(n/2)), where pi is A000720. 16
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. - 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)]; # 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 *)

PROG

(PARI) A056171=n->primepi(n)-primepi(n\2) \\ M. F. Hasler, Dec 31 2016

CROSSREFS

Cf. A001221, A034444, A000720, A048105, A048656, A048657.

Cf. A014085, A060715, A104272, A143223, A143224, A143225, A143226, A143227. [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

Definition simplified by N. J. A. Sloane, Sep 01 2015

STATUS

approved

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Last modified January 17 16:19 EST 2017. Contains 280898 sequences.