A051652 -id:A051652 - cp's OEIS frontend

A051700 Distance from n to closest prime that is different from n.

2, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 3, 2, 1, 2, 1, 2, 1, 2, 3, 2, 1, 4, 1, 2, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 3, 2, 1, 6, 1, 2, 3, 2, 1, 2, 1, 2, 1, 2, 3, 2, 1, 4, 1, 2, 1, 2, 1, 2, 1, 2, 3, 2, 1, 4, 1, 2, 1, 4, 1, 2, 3, 2, 1, 6, 1, 2, 3, 4, 3, 2, 1, 4, 1, 2, 1, 2, 1, 2, 1

Offset: 0

N. J. A. Sloane

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

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

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

with(numtheory); f := n->min(nextprime(n)-n, n-prevprime(n));

Table[Min[NextPrime[n]-n,n-NextPrime[n,-1]],{n,0,200}]  (* Harvey P. Dale, Mar 27 2011 *)

More terms from James Sellers

A046930 Size of sea of composite numbers surrounding n-th prime.

Original entry on oeis.org

1, 1, 2, 4, 4, 4, 4, 4, 8, 6, 6, 8, 4, 4, 8, 10, 6, 6, 8, 4, 6, 8, 8, 12, 10, 4, 4, 4, 4, 16, 16, 8, 6, 10, 10, 6, 10, 8, 8, 10, 6, 10, 10, 4, 4, 12, 22, 14, 4, 4, 8, 6, 10, 14, 10, 10, 6, 6, 8, 4, 10, 22, 16, 4, 4, 16, 18, 14, 10, 4, 8, 12, 12, 10, 8, 8, 12, 10, 10, 16, 10, 10, 10, 6, 8, 8

Offset: 1

N. J. A. Sloane, Marc LeBrun

			23 is in a sea of 8 composites: 20,21,22,23,24,25,26,27,28.

T. D. Noe, Table of n, a(n) for n = 1..10000

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

a046930 1 = 1
a046930 n = subtract 2 $ a031131 n  -- Reinhard Zumkeller, Dec 19 2013

[ seq(ithprime(i)-ithprime(i-2)-2,i=3..100) ];

Table[ Prime[n + 2] - Prime[n] - 2, {n, 75}] (* Robert G. Wilson v Oct 27 2004 *)
Join[{1},#[[3]]-#[[1]]-2&/@Partition[Prime[Range[90]],3,1]] (* Harvey P. Dale, Sep 26 2012 *)

a(n) = A031131(n) - 2 for n > 1. - Reinhard Zumkeller, Dec 19 2013

More terms from Michel ten Voorde

A051701 Closest prime to n-th prime p that is different from p (break ties by taking the smaller prime).

Original entry on oeis.org

3, 2, 3, 5, 13, 11, 19, 17, 19, 31, 29, 41, 43, 41, 43, 47, 61, 59, 71, 73, 71, 83, 79, 83, 101, 103, 101, 109, 107, 109, 131, 127, 139, 137, 151, 149, 151, 167, 163, 167, 181, 179, 193, 191, 199, 197, 199, 227, 229, 227, 229, 241, 239, 257, 251, 257, 271, 269

Offset: 1

N. J. A. Sloane

A227878 gives the terms occurring twice. - Reinhard Zumkeller, Oct 25 2013

			Closest primes to 2,3,5,7,11 are 3,2,3,5,13.

T. D. Noe, Table of n, a(n) for n = 1..1000

Related sequences: A023186, A023187, A023188, A046929, A046930, A046931, A051650, A051652, A051697, A051698, A051699, A051700, A051701, A051702, A051728, A051729, A051730.

Cf. A116946 (semiprime analog).

a051701 n = a051701_list !! (n-1)
a051701_list = f 2 $ 1 : a000040_list where
   f d (q:ps@(p:p':_)) = (if d <= d' then q else p') : f d' ps
     where d' = p' - p
-- Reinhard Zumkeller, Oct 25 2013

a[n_] := (p = Prime[n]; np = NextPrime[p]; pp = NextPrime[p, -1]; If[np-p < p-pp, np, pp]); Table[a[n], {n, 1, 58}] (* Jean-François Alcover, Oct 20 2011 *)
cp[{a_,b_,c_}]:=If[c-bHarvey P. Dale, Oct 08 2012 *)

from sympy import nextprime
def aupton(terms):
  prv, cur, nxt, alst = 0, 2, 3, []
  while len(alst) < terms:
    alst.append(prv if 2*cur - prv <= nxt else nxt)
    prv, cur, nxt = cur, nxt, nextprime(nxt)
  return alst
print(aupton(58)) # Michael S. Branicky, Jun 04 2021

More terms from James Sellers

A051698 Closest prime to n that is different from n (break ties by taking the smaller prime).

Original entry on oeis.org

2, 2, 3, 2, 3, 3, 5, 5, 7, 7, 11, 13, 11, 11, 13, 13, 17, 19, 17, 17, 19, 19, 23, 19, 23, 23, 23, 29, 29, 31, 29, 29, 31, 31, 31, 37, 37, 41, 37, 37, 41, 43, 41, 41, 43, 43, 47, 43, 47, 47, 47, 53, 53, 47, 53, 53, 53, 59, 59, 61, 59, 59, 61, 61, 61, 67, 67, 71, 67, 67, 71, 73

Offset: 0

N. J. A. Sloane

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

Harvey P. Dale, Table of n, a(n) for n = 0..1000

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

cp[n_]:=Module[{p1=NextPrime[n,-1],p2=NextPrime[n]},If[p2-nHarvey P. Dale, Dec 11 2018 *)

More terms from James Sellers

A051729 Smallest number at distance 2n+1 from nearest prime.

Original entry on oeis.org

1, 26, 118, 120, 532, 1140, 1340, 1342, 1344, 15702, 15704, 19632, 19634, 31424, 31426, 31428, 31430, 31432, 155958, 155960, 155962, 155964, 360698, 360700, 370310, 370312, 370314, 370316, 492170, 1349592, 1357262, 1357264, 1357266, 2010800, 2010802, 2010804, 2010806

Offset: 0

N. J. A. Sloane

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

seq[max_] := Module[{s = Table[0, {max}], c = 1, n = 4}, s[[1]] = 1; While[c < max, i = (Min[n - NextPrime[n, -1], NextPrime[n] - n] + 1)/2; If[i <= max && s[[i]] == 0, c++; s[[i]] = n]; n += 2]; s] ; seq[20] (* Amiram Eldar, Aug 28 2021 *)
With[{tbl=Table[{n,If[PrimeQ[n],2,Min[n-NextPrime[n,-1],NextPrime[n]-n]]},{n,500000}]},Table[SelectFirst[tbl,#[[2]]==2k+1&],{k,0,28}]][[;;,1]] (* The program generates the first 29 terms of the sequence. *) (* Harvey P. Dale, Jul 06 2025 *)

a(n) = A051652(2*n+1). - Sean A. Irvine, Oct 01 2021

More terms from James Sellers, Dec 07 1999

More terms from Amiram Eldar, Aug 28 2021

A023187 Distances of increasingly lonely primes to nearest prime.

Original entry on oeis.org

1, 2, 4, 6, 12, 14, 18, 20, 24, 30, 40, 42, 44, 48, 54, 62, 72, 76, 96, 98, 108, 116, 124, 136, 156, 160, 162, 168, 174, 176, 178, 180, 186, 194, 210, 214, 222, 242, 244, 246, 250, 258, 268, 284, 300, 324, 328, 340, 348, 352, 390, 396, 420, 432, 452, 480

Offset: 1

David W. Wilson

nonn

These are the distances mentioned in A023186.

			The nearest prime to 23 is 4 units away, larger than any previous prime, so 4 is in the sequence.

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

t={}; max=p=0; q=2; Do[r=NextPrime[q]; If[(min=Min[q-p,r-q])>max, max=min; AppendTo[t,max]]; p=q; q=r, {n,828000}]; t (* Jayanta Basu, May 18 2013 *)

More terms from Jud McCranie, Jun 16 2000
More terms from T. D. Noe, Jul 21 2006
More terms from Dmitry Petukhov, Oct 03 2015

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

A309877 a(n) is the smallest number k such that the difference between the next prime greater than k and k equals n.

Original entry on oeis.org

1, 0, 8, 7, 24, 23, 90, 89, 118, 117, 116, 115, 114, 113, 526, 525, 524, 523, 888, 887, 1130, 1129, 1338, 1337, 1336, 1335, 1334, 1333, 1332, 1331, 1330, 1329, 1328, 1327, 9552, 9551, 15690, 15689, 15688, 15687, 15686, 15685, 15684, 15683, 19616, 19615, 19614, 19613, 19612, 19611

Offset: 1

Ilya Gutkovskiy, Aug 21 2019

nonn

			+------+------+-----+
| a(n) | next | gap |
|      | prime|     |
+------+------+-----+
|   1  |   2  |  1  |
|   0  |   2  |  2  |
|   8  |  11  |  3  |
|   7  |  11  |  4  |
|  24  |  29  |  5  |
|  23  |  29  |  6  |
|  90  |  97  |  7  |
|  89  |  97  |  8  |
+------+------+-----+

Robert Israel, Table of n, a(n) for n = 1..282

Cf. A000101, A000230, A007918, A007920, A013632, A051652, A075403, A077019, A151800.

N:= 100:
A:= Vector(N,-1):
count:= 0: lastp:= 0:
while count < N do
  p:= nextprime(lastp);
  newvals:= select(t -> A[t]=-1, [$1..min(p-lastp,N)]);
  count:= count+nops(newvals);
  for k in newvals do A[k]:= p-k od;
  lastp:= p;
od:
convert(A,list); # Robert Israel, Aug 23 2019

Table[SelectFirst[Range[0, 20000], NextPrime[#] - # == n &], {n, 1, 50}]
Module[{nn=20000,d},d=Table[{n,NextPrime[n]-n},{n,0,nn}];Table[SelectFirst[d,#[[2]]==k&],{k,50}]][[;;,1]] (* Harvey P. Dale, Mar 23 2025 *)

a(n) = my(k=0); while(nextprime(k+1) - k != n, k++); k; \\ Michel Marcus, Aug 21 2019

a(n) = min {k : A013632(k) = n}.

cp's OEIS Frontend

A051700 Distance from n to closest prime that is different from n.

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A046930 Size of sea of composite numbers surrounding n-th prime.

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A051701 Closest prime to n-th prime p that is different from p (break ties by taking the smaller prime).

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A051698 Closest prime to n that is different from n (break ties by taking the smaller prime).

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A051729 Smallest number at distance 2n+1 from nearest prime.

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A023187 Distances of increasingly lonely primes to nearest prime.

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

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A309877 a(n) is the smallest number k such that the difference between the next prime greater than k and k equals n.