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A056171
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a(n) = pi(n) - pi(floor(n/2)), where pi is A000720.
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25
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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
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OFFSET
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1,3
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COMMENTS
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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
First occurrence of k is at n = A080359(k).
The last occurrence of k is at n = A080360(k).
The number of times k appears is A080362(k).
(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
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LINKS
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FORMULA
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EXAMPLE
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10! = 2^8 * 3^2 * 5^2 * 7. The only unitary prime divisor is 7, so a(10)=1.
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MAPLE
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numtheory[pi](x)-numtheory[pi](floor(x/2)) ;
end proc:
A056171 := n -> nops(select(isprime, [$iquo(n, 2)+1..n])):
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MATHEMATICA
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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 *)
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PROG
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(Python)
from sympy import primepi
[primepi(n) - primepi(n//2) for n in range(1, 151)] # Indranil Ghosh, Mar 22 2017
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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