A051699 Distance from n to closest prime.
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
Offset: 0
Examples
Closest primes to 0,1,2,3,4 are 2,2,2,3,3.
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
Crossrefs
Programs
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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
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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-François Alcover, Apr 05 2012 *) -
PARI
a(n)=if(n<1,2*(n==0),vecmin(vector(n,k,abs(n-prime(k)))))
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PARI
a(n)=if(n<1,2*(n==0),min(nextprime(n)-n,n-precprime(n)))
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Python
from sympy import prevprime, nextprime, isprime def A051699(n): return nextprime(n) - n if n <= 1 else 0 if isprime(n) else min(n-prevprime(n), nextprime(n)-n) # Ya-Ping Lu, Mar 22 2025
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
C = lim_{n->oo} S(n)/(n*log(n)) = 0.44 approximately. - Ya-Ping Lu, Apr 06 2025
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.
abs(a(n)-a(n+1)) = 1 if n != 2; a((p+q)/2 +- k) = (q-p)/2 - k, where p < q are two consecutive primes and k = 0, 1, 2, ..., (q-p)/2. - Ya-Ping Lu, Mar 22 2025
Extensions
More terms from James Sellers