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 A056171 a(n) = pi(n) - pi(floor(n/2)), where pi is A000720. 21
 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 times k appears: 4, 10, 7, 14, 7, 10, 12, 5, 14, 16, 3, 10, 18, 16, 15, ..., . (End) Lev Schnirelmann proved that for every n, a(n) > (1/log_2(n))*(n/3 - 4*sqrt(n)) - 1 - (3/2)*log_2(n). - Arkadiusz Wesolowski, Nov 03 2017 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! = 2^8 * 3^2 * 5^2 * 7. The only unitary prime divisor is 7, so a(10)=1. MAPLE A056171 := proc(x)      numtheory[pi](x)-numtheory[pi](floor(x/2)) ; end proc: seq(A056171(n), 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 [primepi(n) - primepi(n//2) for n in range(1, 151)] # Indranil Ghosh, Mar 22 2017 CROSSREFS Cf. A001221, A034444, A000720, A048105, A048656, A048657. Cf. A014085, A060715, A104272, A143223, A143224, A143225, A143226, A143227. Cf. A080359, A080360. - Robert G. Wilson v, Mar 20 2017 Sequence in context: A316112 A317994 A128428 * A333749 A238949 A076755 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 9 13:42 EDT 2021. Contains 343742 sequences. (Running on oeis4.)