A051697 -id:A051697 - cp's OEIS frontend

A046929 Width of moat of composite numbers surrounding n-th prime.

0, 0, 1, 1, 1, 1, 1, 1, 3, 1, 1, 3, 1, 1, 3, 5, 1, 1, 3, 1, 1, 3, 3, 5, 3, 1, 1, 1, 1, 3, 3, 3, 1, 1, 1, 1, 5, 3, 3, 5, 1, 1, 1, 1, 1, 1, 11, 3, 1, 1, 3, 1, 1, 5, 5, 5, 1, 1, 3, 1, 1, 9, 3, 1, 1, 3, 5, 5, 1, 1, 3, 5, 5, 5, 3, 3, 5, 3, 3, 7, 1, 1, 1, 1, 3, 3, 5, 3, 1, 1, 3, 7, 3, 3

Offset: 1

N. J. A. Sloane

			23 has a buffer of 3 composites around it on each side: 20,21,22,23,24,25,26.

T. D. Noe, Table of n, a(n) for n = 1..10000
Jean-François Alcover, Frequency distribution up to 10^6 terms.

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

with(numtheory); a := i->min(ithprime(n)-ithprime(n-1)-1, ithprime(n+1)-ithprime(n)-1);

a[n_] := Min[Prime[n] - Prime[n-1] - 1, Prime[n+1] - Prime[n] - 1]; a[1] = 0; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Apr 16 2013 *)
Join[{0},Min[Differences[#]]&/@Partition[Prime[Range[100]],3,1]-1] (* Harvey P. Dale, Feb 24 2014 *)

A051650 Lonely numbers: distance to closest prime sets a new record.

Original entry on oeis.org

0, 23, 53, 120, 211, 1340, 1341, 1342, 1343, 1344, 2179, 3967, 15704, 15705, 16033, 19634, 19635, 24281, 31428, 31429, 31430, 31431, 31432, 31433, 38501, 58831, 155964, 203713, 206699, 370310, 370311, 370312, 370313, 370314, 370315, 370316

Offset: 0

N. J. A. Sloane

			23 is 4 units away from the closest prime (not including itself), so 23 is in the sequence.

Charles R Greathouse IV and Giovanni Resta, Table of n, a(n) for n = 0..211 (terms < 10^14, first 156 terms from Charles R Greathouse IV)

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

Distances are in A051730.

d[0] = 2; d[k_] := Min[k - NextPrime[k, -1], NextPrime[k] - k]; a[0] = 0; a[n_] := a[n] = (k = a[n-1] + 1; record = d[a[n-1]]; While[d[k] <= record, k++]; k); Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Jan 16 2012 *)
dcp[n_]:=Min[n-NextPrime[n,-1],NextPrime[n]-n]; DeleteDuplicates[Table[{n,dcp[n]},{n,0,375000}],GreaterEqual[#1[[2]],#2[[2]]]&][[;;,1]] (* Harvey P. Dale, Feb 23 2023 *)

print1(0);w=2;p=2;q=3;forprime(r=5,1e9,if(p+w+ww,w=t;print1(", "q));p=q;q=r) \\ Charles R Greathouse IV, Jan 16 2012

More terms from James Sellers, Dec 23 1999 and from Jud McCranie, Jun 16 2000

A051730 Distance from A051650(n) to nearest prime.

Original entry on oeis.org

2, 4, 6, 7, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 24, 25, 26, 30, 31, 32, 33, 34, 35, 36, 40, 42, 43, 44, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 96, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109

Offset: 0

N. J. A. Sloane

			23 is 4 units away from the closest prime (not including itself), so 4 is in the sequence.

Amiram Eldar, Table of n, a(n) for n = 0..211 (calculated from the b-file at A051650)

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

(* b stands for A051650 *) d[0] = 2; d[k_] := Min[k - NextPrime[k, -1], NextPrime[k] - k]; b[0] = 0; b[n_] := b[n] = (k = b[n-1] + 1; record = d[b[n-1]]; While[d[k] <= record, k++]; k); a[n_] := a[n] = d[b[n]]; Table[ Print[ a[n]]; a[n], {n, 0, 66}] (* Jean-François Alcover, Jan 16 2012 *)

print1(w=2);p=2;q=3;forprime(r=5,1e9,if(p+w+ww,w=t;print1(", "t));p=q;q=r) \\ Charles R Greathouse IV, Jan 16 2012

[10] C#=pack(3,5):R=2:N=4:print 2; [20] if N>member(C#,2) then C#=pack(member( C#,2)):C#=C#+nxtprm(member(C#,1)) [30] Prv=member(C#,1):Nxt=member(C#,2) [40] if Nxt=N then Nxt=nxtprm(N) [50] if (N-Prv)>=(Nxt-N) then P=Nxt-N else P=N-Prv [60] if P>R then print P;:R=P [70] N+=1 :goto 20

More terms from James Sellers, Dec 23 1999 and from Jud McCranie, Jun 16 2000

Further terms from Naohiro Nomoto, Jun 21 2001

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

Original entry on oeis.org

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

A077018 Closest prime to n (break ties by taking the larger prime).

Original entry on oeis.org

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

Offset: 0

Eric W. Weisstein, Oct 17 2002

nonn

Eric Weisstein's World of Mathematics, Nearest Prime

This is the same as A051697 except that one of the 3s is deleted here.

a[n_] := (np = NextPrime[n]; pp = NextPrime[np, -1]; Which[np > 2n - pp, pp, np < 2n - pp, np, True, np]); a[0] = a[1] = 2; Table[a[n], {n, 0, 71}] (* Jean-François Alcover, Oct 31 2012 *)

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

A046929 Width of moat of composite numbers surrounding n-th prime.

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A051650 Lonely numbers: distance to closest prime sets a new record.

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A051730 Distance from A051650(n) to nearest prime.

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UBASIC

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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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