0,1

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

Eric Weisstein's World of Mathematics, Prime Distance

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

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

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

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)))))

(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

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

nonn,easy,nice

More terms from James Sellers

approved