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

0,1

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

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

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-François 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)))
(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

Crossrefs

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

Sequence in context: A333816 A363709 A246398 * A328682 A007920 A127587

Adjacent sequences: A051696 A051697 A051698 * A051700 A051701 A051702

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Extensions

More terms from James Sellers

Status

approved