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A051699 Distance from n to closest prime. 22

%I

%S 2,1,0,0,1,0,1,0,1,2,1,0,1,0,1,2,1,0,1,0,1,2,1,0,1,2,3,2,1,0,1,0,1,2,

%T 3,2,1,0,1,2,1,0,1,0,1,2,1,0,1,2,3,2,1,0,1,2,3,2,1,0,1,0,1,2,3,2,1,0,

%U 1,2,1,0,1,0,1,2,3,2,1,0,1,2,1,0,1,2,3,2,1,0,1,2,3,4,3,2,1,0,1,2,1,0,1,0,1

%N Distance from n to closest prime.

%H T. D. Noe, <a href="/A051699/b051699.txt">Table of n, a(n) for n = 0..10000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PrimeDistance.html">Prime Distance</a>

%H <a href="/index/Di#distance_to_the_nearest">Index entries for sequences related to distance to nearest element of some set</a>

%F Conjecture: S(n) = sum(k=1, n, a(k) ) is asymptotic to C*n*log(n) with C=0.29...... - _Benoit Cloitre_, Aug 11 2002

%F Comment from _Giorgio Balzarotti_, Sep 18 2005: by means of the Prime Number Theorem is possible to derive the following inequality : c1*n*log(n) < S(n)< c2*n*log(n), where log is the natural logarithm and c1 = 1/4 and c2 = 3/8 (for n > 130). For a more accurate estimation of the values for c1 and c2, it necessary to know the number of twin primes with respect to the total number of primes.

%e Closest primes to 0,1,2,3,4 are 2,2,2,3,3.

%p A051699 := proc(n) if isprime(n) then 0; elif n<= 2 then 2-n ; else min(nextprime(n)-n, n-prevprime(n)) ; end if ; end proc; # _R. J. Mathar_, Nov 01 2009

%t FormatSequence[ Table[Min[Abs[n-If[n<2, 2, Prime[{#, #+1}&[PrimePi[n]]]]]], {n, 0, 101}], 51699, 0, Name->"Distance to closest prime." ]

%t (* From version 6 on: *) a[_?PrimeQ] = 0; a[n_] := Min[NextPrime[n]-n, n-NextPrime[n, -1]]; Table[a[n], {n, 0, 104}] (* _Jean-Fran├žois Alcover_, Apr 05 2012 *)

%o (PARI) a(n)=if(n<1,2*(n==0),vecmin(vector(n,k,abs(n-prime(k)))))

%o (PARI) a(n)=if(n<1,2*(n==0),min(nextprime(n)-n,n-precprime(n)))

%Y Related sequences: A023186-A023188, A046929-A046931, A051650, A051652, A051697-A051702, A051728-A051730.

%K nonn,easy,nice

%O 0,1

%A _N. J. A. Sloane_

%E More terms from _James A. Sellers_

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Last modified August 22 07:40 EDT 2019. Contains 326172 sequences. (Running on oeis4.)