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A063124 a(n) = # { primes p | prime(n) <= p < 2*prime(n) } where prime(n) is the n-th prime.

%I A063124 #48 Dec 12 2024 10:55:59
%S A063124 2,2,2,3,4,4,5,5,6,7,8,10,10,10,10,12,14,13,14,15,14,16,16,17,20,21,
%T A063124 20,20,19,19,24,24,26,26,28,27,29,29,29,29,31,31,33,33,33,33,36,39,39,
%U A063124 39,40,40,40,42,43,44,43,43,43,43,43,45,50,51,50,50,55,55,57,56,56,56,58
%N A063124 a(n) = # { primes p | prime(n) <= p < 2*prime(n) } where prime(n) is the n-th prime.
%C A063124 a(n) is the number of primes between prime(n) and 2*prime(n) inclusive. - _Sean A. Irvine_, Apr 18 2023
%C A063124 Also for x = Product_{i=n..n+k} A000040(i), the least k such that A003961(x) > 2*x. - _Antti Karttunen_, Dec 08 2024
%H A063124 N. J. A. Sloane, <a href="/A063124/b063124.txt">Table of n, a(n) for n = 1..20000</a> [First 2000 terms from Harry J. Smith]
%F A063124 a(n) = A035250(prime(n)).
%F A063124 a(n) = A070046(n) + 1. - _Sean A. Irvine_, Apr 18 2023
%F A063124 From _Antti Karttunen_, Dec 08 2024: (Start)
%F A063124 a(n) = n-A331677(n) = 1+n-A334051(n).
%F A063124 a(n) = 1+A000720(2*A000040(n))-n. [After _Harry J. Smith_'s PARI-program]
%F A063124 a(n) < A108227(n). [Assuming _M. F. Hasler_'s interpretation in May 08 2017 comment in the latter]
%F A063124 a(n) = A001222(A378746(n)).
%F A063124 (End)
%e A063124 a(10) = 7 as there are 7 primes between prime(10) = 29 and 58 = 29*2: 29, 31, 37, 41, 43, 47, 53.
%p A063124 A062134 := proc(n) numtheory:-pi(2*ithprime(n))-n+1; end; # _N. J. A. Sloane_, Oct 19 2024
%p A063124 [seq(A062134(n),n=1..100)];
%t A063124 Table[PrimePi[2*Prime[n]] - n + 1, {n, 100}] (* _Paolo Xausa_, Oct 22 2024 *)
%o A063124 (PARI) a(n)={1 + primepi(2*prime(n)) - n} \\ _Harry J. Smith_, Aug 19 2009
%Y A063124 Related sequences:
%Y A063124 Primes (p) and composites (c): A000040, A002808, A000720, A065855.
%Y A063124 Primes between p(n) and 2*p(n): A063124, A070046; between c(n) and 2*c(n): A376761; between n and 2*n: A035250, A060715, A077463, A108954.
%Y A063124 Composites between p(n) and 2*p(n): A246514; between c(n) and 2*c(n): A376760; between n and 2*n: A075084, A307912, A307989, A376759.
%Y A063124 Cf. A003961, A108227, A331677, A334051, A378746.
%K A063124 nonn
%O A063124 1,1
%A A063124 _Reinhard Zumkeller_, Aug 08 2001
%E A063124 Definition clarified by _N. J. A. Sloane_, Oct 04 2024