A023186 -id:A023186 - 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

A102723 Smallest prime a(n) such that a(n)-x and a(n)+x, for x=1 to n, are all composite.

Original entry on oeis.org

5, 23, 23, 53, 53, 211, 211, 211, 211, 211, 211, 1847, 1847, 2179, 2179, 2179, 2179, 3967, 3967, 16033, 16033, 16033, 16033, 24281, 24281, 24281, 24281, 24281, 24281, 38501, 38501, 38501, 38501, 38501, 38501, 38501, 38501, 38501, 38501, 58831

Offset: 1

Ray G. Opao, Feb 06 2005

nonn

a(2n+1)=a(2n). - Robert G. Wilson v, Feb 22 2005

Using Dirichlet's theorem, Sierpiński (1948) proved that a(n) exists for all n > 0. He noted that a(n) is a non-twin prime (A007510), except for a(1) = 5. - Jonathan Sondow, Oct 27 2017

David A. Corneth, Table of n, a(n) for n = 1..479 (first 97 terms from Harvey P. Dale)
W. Sierpiński, Remarque sur la répartition des nombres premiers, Colloq. Math., 1 (1948), 193-194.

Cf. A007510, A023186.

f[n_] := Block[{k = 1}, While[ Union[ PrimeQ /@ Sort[ Flatten[ Table[{Prime[k] - i, Prime[k] + i}, {i, n}]]]] != {False}, k++ ]; Prime[k]]; Table[ f[n], {n, 40}] (* Robert G. Wilson v, Feb 22 2005 *)
cmpgap[n_]:=Module[{p=Prime[n]},Min[p-NextPrime[p,-1],NextPrime[p]-p]]; Module[{nn=10000,prs},prs=Table[{Prime[n],cmpgap[n]},{n,nn}];Table[ SelectFirst[ prs,#[[2]]>=k&],{k,2,50}]][[All,1]] (* Harvey P. Dale, Oct 15 2021 *)

a(12)-a(40) from Robert G. Wilson v, Feb 22 2005

A330428 Smallest prime p such that both nearest primes up and down are farther away than n*log(p).

Original entry on oeis.org

5, 211, 38501, 413353, 10938023, 142414669, 163710121, 8835528511, 31587561361, 343834606051, 1480975873513, 26923643849953

Offset: 1

Steven M. Altschuld, Dec 14 2019

For these numbers, the name "Lowest Frogger Primes" LFP(n) is suggested because (frog) jumps with a length greater than n times the local average are required to bridge the gaps (logs).

Cf. A288908 (with 1*log(P)), A330426 (with 2*log(P)), A330427 (with 3*log(P)).

Cf. A023186, A120937.

{my(npp=2,np=3,n=1);forprime(p=5,10^9,my(d=log(p)*n);if(np-npp>d&&p-np>d,print(np,", ");n++);npp=np;np=p)} \\ Hugo Pfoertner, Dec 14 2019

a(5)-a(9) from Hugo Pfoertner, Dec 14 2019
a(10) from Hugo Pfoertner, Dec 16 2019
a(11)-a(12) from Giovanni Resta, Dec 19 2019

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

A087770 "Lonely primes": those primes that are locally maximally isolated from the nearest other primes. The differences between each lonely prime and the immediately preceding prime and following primes are both greater than the corresponding differences for all lonely primes earlier in the sequence.

Original entry on oeis.org

2, 3, 7, 23, 89, 211, 1847, 2179, 14107, 33247, 38501, 58831, 268343, 1272749, 2198981, 10938023, 72546283, 162821917, 325737821, 2888688863, 6613941601, 11179888193, 24016237123, 96155166493, 179474021633, 215686840471, 633880576177, 1480975873513, 9156364643509

Offset: 1

Walter Carlini, Oct 03 2003

nonn

The concept of "lonely prime" is similar to that of maximal prime gaps since lonely primes are increasingly distant from each other.

See A023186 for another version of this sequence, which only requires increasing the minimum of the two gaps to the neighbors. The definition from A023186 seems to be the more common variant. - Hugo Pfoertner, Dec 17 2019

			a(0) = 2.
a(1) = 3 because 3 - 2 = 1 and 5 - 3 = 2.
a(2) = 7 because 7 - 5 = 2 (and 2 > 3 - 2) and 11 - 7 = 4 (and 4 > 5 - 3).
a(3) = 23 because 23 - 19 = 4 ( 23 - 19 > 7 - 5) and 29 - 23 = 6 (29 - 23 > 11 - 7).
a(4) = 89 because 89 - 83 = 6 > 23 - 19 and 97 - 89 = 8 > 29 - 23.
Note, for example, that 53 is not a lonely prime because 53 - 47 = 6, which is > 23 - 19 however 59 - 53 = 6, which is not > 29 - 23.

Cf. A000230, A002386, A023186.

NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; p = 2; q = 2; r = 3; d = e = 0; Do[ While[ q - p <= d || r - q <= e, p = q; q = r; r = NextPrim[r]]; Print[q]; d = Max[q - p, d]; e = Max[r - q, e]; p = q; q = r; r = NextPrim[r], {n, 1, 40}] (* Robert G. Wilson v *)

Corrected and extended by Ray Chandler, Oct 06 2003
Offset changed and a(21)-a(27) from Hugo Pfoertner, Dec 17 2019
a(28)-a(29) from Giovanni Resta, Dec 17 2019

A120937 Least prime such that the distance to the two adjacent primes is 2n or greater.

Original entry on oeis.org

3, 5, 23, 53, 211, 211, 211, 1847, 2179, 2179, 3967, 16033, 16033, 24281, 24281, 24281, 38501, 38501, 38501, 38501, 38501, 58831, 203713, 206699, 206699, 413353, 413353, 413353, 1272749, 1272749, 1272749, 1272749, 2198981, 2198981, 2198981

Offset: 0

T. D. Noe, Jul 21 2006

nonn

Erdos and Suranyi call these reclusive primes and prove that such a prime exists for all n. Except for a(0), the record values are in A023186.

			a(3)=53 because the adjacent primes 47 and 59 are at distance 6 and all smaller primes have a closer distance.

Paul Erdős and Janos Suranyi, Topics in the theory of numbers, Springer, 2003.

Cf. A023186, A087770, A330428.

k=2; Table[While[Prime[k]-Prime[k-1]<2n || Prime[k+1]-Prime[k]<2n, k++ ]; Prime[k], {n,0,40}]

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

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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A102723 Smallest prime a(n) such that a(n)-x and a(n)+x, for x=1 to n, are all composite.

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A330428 Smallest prime p such that both nearest primes up and down are farther away than n*log(p).

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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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