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A051699 Distance from n to closest prime. 22
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, 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, 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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

LINKS

T. D. Noe, Table of n, a(n) for n = 0..10000

Eric Weisstein's World of Mathematics, Prime Distance

Index entries for sequences related to distance to nearest element of some set

FORMULA

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

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.

EXAMPLE

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

MAPLE

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

MATHEMATICA

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

(* 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 *)

PROG

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

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

CROSSREFS

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

Sequence in context: A130068 A316825 A246398 * A007920 A127587 A175832

Adjacent sequences:  A051696 A051697 A051698 * A051700 A051701 A051702

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from James A. Sellers

STATUS

approved

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Last modified July 17 14:44 EDT 2019. Contains 325106 sequences. (Running on oeis4.)