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%I #23 Mar 18 2019 08:10:23
%S 0,1,1,1,2,1,1,1,2,2,2,2,3,2,2,3,3,2,2,2,3,3,3,2,3,3,3,4,5,4,4,4,4,4,
%T 4,5,5,4,4,5,6,5,5,4,5,5,5,5,6,6,6,6,7,6,6,7,7,7,7,7,7,6,6,6,7,7,7,7,
%U 8,8,8,8,9,8,8,9,9,9,9,8,8,8,8,7,8,8,8,9,9,8,8,9,10,10,10
%N Number of primes in the half-open interval [n, 3*n/2).
%C Suggested by Bertrand's postulate (actually a theorem): for all x > 1, there is a prime number between x and 2x (see related references and links mentioned in A060715, A166968 and A143227).
%C For all n > 1, a(n)>=1 (that is, there is always a prime between n and 3*n/2); this can be seen using the stronger result proved by Jitsuro Nagura in 1952: for n >= 25, there is always a prime between n and (1 + 1/5)n.
%C Successive terms vary by no more than +/- one unit.
%H Jean-Christophe Hervé, <a href="/A230772/b230772.txt">Table of n, a(n) for n = 1..6667</a>
%H Jitsuro Nagura, <a href="https://doi.org/10.3792/pja/1195570997">On the interval containing at least one prime number</a>, Proc. Japan Acad. 28, 1952, 177-181.
%F a(n) = sum(A010051(n+k): 0<=k<3*n/2).
%F a(n) = A000720(ceiling(1.5*n)-1) - A000720(n-1).
%e a(29)=5 since five primes (29,31,37,41,43) are located between 29 and 43.5.
%t a[n_]:=PrimePi[Ceiling[1.5*n]-1]-PrimePi[n-1]; Table[a[n], {n, 2, 100}]
%o (R)
%o nvalues <- 1000
%o A <- vector('numeric', nvalues)
%o A[1] <- 0
%o # primepi = table of the primepi sequence A000720
%o for(i in 2:nvalues) A[i] <- primepi[ceiling(1.5*i)-1]-primepi[i-1]
%Y Cf. A060715, A166968, A143227, A000720, A010051.
%K nonn,easy
%O 1,5
%A _Jean-Christophe Hervé_, Oct 30 2013
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