A132470 -id:A132470 - cp's OEIS frontend

A051728 Smallest number at distance 2n from nearest prime.

2, 0, 23, 53, 409, 293, 211, 1341, 1343, 2179, 3967, 15705, 16033, 19635, 31425, 24281, 31429, 31431, 31433, 155959, 38501, 58831, 203713, 268343, 206699, 370311, 370313, 370315, 370317, 1349591, 1357261, 1272749, 1357265, 1357267, 2010801, 2010803, 2010805, 2010807

Offset: 0

N. J. A. Sloane

a(0) = 2. For n > 0, let f(m) = minimal distance from m to closest prime (excluding m itself). The a(n) = min { m : f(m) = 2n }.

f(m) is tabulated in A051700. - R. J. Mathar, Nov 18 2007

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

Cf. A132470.

A051700 := proc(m) if m <= 2 then op(m+1,[2,1,1]) ; else min(nextprime(m)-m,m-prevprime(m)) ; fi ; end: A051728 := proc(n) local m ; if n = 0 then RETURN(2); else for m from 0 do if A051700(m) = 2 * n then RETURN(m) ; fi ; od: fi ; end: seq(A051728(n),n=0..20) ; # R. J. Mathar, Nov 18 2007

a[n_] := Module[{m}, If[n == 0, Return[2], For[m = 0, True, m++, If[Min[NextPrime[m]-m, m-NextPrime[m, -1]] == 2*n, Return[m]]]]]; Table[Print[an = a[n]]; an, {n, 0, 33}] (* Jean-François Alcover, Feb 11 2014, after R. J. Mathar *)
Join[{2},With[{t=Table[{n,Min[n-NextPrime[n,-1],NextPrime[n]-n]},{n,0,1358000}]},Table[SelectFirst[t,#[[2]]==2k&],{k,33}]][[All,1]]] (* Harvey P. Dale, Aug 13 2019 *)

a(n) = A051652(2*n). - Sean A. Irvine, Oct 01 2021

More terms from James Sellers, Dec 07 1999

More terms from Amiram Eldar, Aug 28 2021

A132861 Smallest number at distance 3n from nearest prime (variant 2).

Original entry on oeis.org

2, 26, 53, 532, 211, 1342, 2179, 15704, 16033, 31424, 24281, 31430, 31433, 155960, 58831, 360698, 206699, 370312, 370315, 492170, 1357261, 1357264, 1357267, 2010802, 2010805, 4652428, 12485141, 17051788, 17051791, 17051794, 11117213, 20831416, 10938023, 20831422

Offset: 0

R. J. Mathar, Nov 18 2007

nonn

Let f(m) be the distance to the nearest prime as defined in A051700(m). Then a(n) = min {m: f(m) = 3n} for n > 0. A132470 uses A051699(m) to define the distance. a(n) <= A132470(n) because here primes at the start or end of a prime gap of size 3n may be picked, which would be discarded in A132470 for n>0; this gives a chance to minimize m here further than in A132470.

Michael S. Branicky, Table of n, a(n) for n = 0..76
Michael S. Branicky, Python program

Cf. A132470, A051700.

A051700 := proc(m) if m <= 2 then op(m+1,[2,1,1]) ; else min(nextprime(m)-m,m-prevprime(m)) ; fi ; end: a := proc(n) local m ; if n = 0 then RETURN(2); else for m from 0 do if A051700(m) = 3 * n then RETURN(m) ; fi ; od: fi ; end: seq(a(n),n=0..18);

# see link for faster program
from sympy import prevprime, nextprime
def A051700(n):
  return [2, 1, 1][n] if n < 3 else min(n-prevprime(n), nextprime(n)-n)
def a(n):
  if n == 0: return 2
  m = 0
  while A051700(m) != 3*n: m += 1
  return m
print([a(n) for n in range(13)]) # Michael S. Branicky, Feb 26 2021

a(n) = min {m : A051700(m) = 3n} for n > 0.

a(n) = A051652(3*n). [From R. J. Mathar, Jul 22 2009]

7 more terms from R. J. Mathar, Jul 22 2009
4 more terms from R. J. Mathar, Aug 21 2018
a(30) and beyond and edits from Michael S. Branicky, Feb 26 2021

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A051728 Smallest number at distance 2n from nearest prime.

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