A095117 - OEIS

%I #42 Feb 16 2024 10:15:20

%S 0,1,3,5,6,8,9,11,12,13,14,16,17,19,20,21,22,24,25,27,28,29,30,32,33,

%T 34,35,36,37,39,40,42,43,44,45,46,47,49,50,51,52,54,55,57,58,59,60,62,

%U 63,64,65,66,67,69,70,71,72,73,74,76,77,79,80,81,82,83,84,86,87,88,89,91

%N a(n) = pi(n) + n, where pi(n) = A000720(n) is the number of primes <= n.

%C Positions of first occurrences of n in A165634: A165634(a(n))=n for n>0. - _Reinhard Zumkeller_, Sep 23 2009

%C There exists at least one prime number p such that n < p <= a(n) for n >= 2. For example, 2 is in (2, 3], 5 in (3, 5], 5 in (4, 6], ..., and primes 73, 79, 83 and 89 are in (71, 91] (see Corollary 1 in the paper by Ya-Ping Lu attached in the links section). - _Ya-Ping Lu_, Feb 21 2021

%H Carmine Suriano, <a href="/A095117/b095117.txt">Table of n, a(n) for n = 0..9999</a>

%H Ya-Ping Lu, <a href="/A337788/a337788.pdf">Lower Bounds for the Number of Primes in Some Integer Intervals</a>

%F a(0) = 0; for n>0, a(n) = a(n-1) + (if n is prime then 2, else 1). - _Robert G. Wilson v_, Apr 22 2007; corrected by _David James Sycamore_, Aug 16 2018

%p with(numtheory): seq(n+pi(n),n=1..90); # _Emeric Deutsch_, May 02 2007

%t Table[ PrimePi@n + n, {n, 0, 71}] (* _Robert G. Wilson v_, Apr 22 2007 *)

%o (Haskell)

%o a095117 n = a000720 n + toInteger n  -- _Reinhard Zumkeller_, Apr 17 2012

%o (PARI) a(n) = n + primepi(n); \\ _Michel Marcus_, Feb 21 2021

%o (Python)

%o from sympy import primepi

%o def a(n): return primepi(n) + n

%o print([a(n) for n in range(72)]) # _Michael S. Branicky_, Feb 21 2021

%Y Complement of A095116.

%Y Cf. A000720, A064427.

%K easy,nonn

%O 0,3

%A _Dean Hickerson_, following a suggestion of _Leroy Quet_, May 28 2004

%E Edited by _N. J. A. Sloane_, Jul 02 2008 at the suggestion of _R. J. Mathar_