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A056171 a(n) = pi(n) - pi(floor(n/2)), where pi is A000720. 17
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

From Robert G. Wilson v, Mar 20 2017: (Start)

First occurrence of k: 2, 3, 13, 19, 31, 43, 53, 61, 71, 73, 101, 103, 109, 113, 139, 157, 173, 181, 191, 193, ...,  = A080359.

The last occurrence of k: 10, 16, 28, 40, 46, 58, 66, 70, 96, 100, 106, 126, 148, 150, 166, 178, 180, 226, 228, ..., = A080360.

The number of terms in the factorization of n! is A000720.

The number of times k appears: 4, 10, 7, 14, 7, 10, 12, 5, 14, 16, 3, 10, 18, 16, 15, ..., .

(End)

LINKS

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

FORMULA

a(n) = A000720(n) - A056172(n). - Robert G. Wilson v, Apr 09 2017

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

(Python)

from sympy import primepi

print [primepi(n) - primepi(n/2) for n in xrange (1, 151)] # Indranil Ghosh, Mar 22 2017

CROSSREFS

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

Cf. A014085, A060715, A104272, A143223, A143224, A143225, A143226, A143227. [Jonathan Sondow, Aug 03 2008]

Cf. A080359, A080360. - Robert G. Wilson v, Mar 20 2017

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 May 24 11:40 EDT 2017. Contains 286975 sequences.