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A051699
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Distance from n to closest prime.
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28
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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
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OFFSET
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0,1
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LINKS
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FORMULA
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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 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.
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EXAMPLE
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Closest primes to 0,1,2,3,4 are 2,2,2,3,3.
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MAPLE
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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
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MATHEMATICA
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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-François Alcover, Apr 05 2012 *)
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PROG
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(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)))
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CROSSREFS
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KEYWORD
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nonn,easy,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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