A046931 -id:A046931 - cp's OEIS frontend

A051730 Distance from A051650(n) to nearest prime.

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

A261525 a(n) = smallest m such that A031131(m) = 2*n.

Original entry on oeis.org

2, 3, 9, 8, 15, 23, 47, 29, 66, 114, 46, 220, 188, 258, 640, 375, 480, 589, 216, 326, 367, 1006, 738, 1183, 1985, 1847, 1662, 2224, 3731, 3861, 3561, 2699, 3792, 4521, 2225, 12541, 3384, 12761, 3385, 4058, 10228, 15747, 15927, 14357, 18280, 19025, 14123

Offset: 2

Reinhard Zumkeller, Aug 23 2015

nonn

A031131(a(n)) = 2*n and A031131(m) != 2*n for m < a(n);
A046931(n) = A000040(a(n)+1);
a(n)-th and (a(n)+2)-nd primes are the first pair that differ by 2*n;
conjecture: sequence is defined for all n > 1.

Reinhard Zumkeller, Table of n, a(n) for n = 2..100

Cf. A031131, A038664, A046931.

a261525 = (+ 1) . fromJust . (`elemIndex` a031131_list) . (* 2)

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

A098968 Record values for size of sea of composite numbers surrounding primes.

Original entry on oeis.org

0, 1, 2, 4, 8, 10, 12, 16, 22, 26, 38, 40, 42, 46, 48, 54, 56, 70, 74, 78, 80, 98, 112, 114, 124, 136, 138, 158, 188, 198, 232, 234, 244, 246, 268, 304, 326, 328, 334, 338, 370

Offset: 0

N. J. A. Sloane, Oct 24 2004

nonn

For a prime p let s(p) (essentially A046930) denote the number of composite numbers less than p and greater than max{1, previous prime} or greater than p and less than the next prime. Sequence gives record values of s(p).

Records in A046930 (if initial term is 0 not 1). Cf. A098969, A046931.

a = {1, 2}; b = {0, 1}; d = 1; p = 2; q = 3; Do[ r = Prime[n]; c = r - p - 2; If[c > d, Print[{n + 1, c}]; d = c; AppendTo[a, n - 1]; AppendTo[b, c]]; p = q; q = r, {n, 3, 10^9}]; b (* Robert G. Wilson v, Oct 27 2004 *)
Join[{0},DeleteDuplicates[Total/@Partition[Differences[Prime[Range[5*10^6]]]-1,2,1],GreaterEqual]] (* The program generates the first 34 terms of the sequence. *) (* Harvey P. Dale, May 25 2025 *)

More terms from Robert G. Wilson v and Olaf Voß, Oct 27 2004

A377971 Square array of primes p >= 7, read by decreasing antidiagonals. Each row lists, in increasing order, the primes that share the same sum of their neighboring prime gaps.

Original entry on oeis.org

7, 11, 29, 13, 31, 23, 17, 59, 37, 53, 19, 61, 47, 97, 89, 41, 73, 67, 139, 199, 223, 43, 137, 79, 149, 359, 251, 113, 71, 151, 83, 157, 367, 337, 127, 331, 101, 179, 131, 173, 389, 467, 307, 479, 631, 103, 239, 163, 181, 449, 547, 317, 523, 797, 211, 107, 269, 167, 191, 521, 557, 409, 953, 1087, 293, 1381

Offset: 1

Tamas Sandor Nagy, Nov 13 2024

First column is subset of A046931, which starts with 3. Here, 3 and 5 are omitted.

The related sum can be denoted Sum_prime_gaps, S = pg_inf + pg_sup.

			Square array begins:
.
S = pg_inf + pg_sup |
      2*(3..k)      |
-----------------------------------------------------------------------
          6         |   7,  11,  13,  17,  19,  41,  43,  71, 101, ... A098414
          8         |  29,  31,  59,  61,  73, 137, 151, 179, 239, ...
         10         |  23,  37,  47,  67,  79,  83, 131, 163, 167, ...
         12         |  53,  97, 139, 149, 157, 173, 181, 191, 241, ...
         14         |  89, 199, 359, 367, 389, 449, 521, 619, 661, ...
.
31, 59 and 179 are in the same row because their preceding and succeeding prime gaps, (pg_inf, pg_sup), respectively (2,6), (6,2) and (6,2) each equally sum up to 8.
53 and 181 are in the same row because their preceding and succeeding prime gaps, (pg_inf, pg_sup), respectively (6,6) and (2,10) each equally sum up to 12. Here, 53 also happens to be a balanced prime as its corresponding gaps, (6,6), are equal.

Cf. A000040, A001223, A098414, A002260, A006562.

Sum_prime_gaps_a(n) = S_a(n) = (A002260(n))*2 + 4.

cp's OEIS Frontend

A051730 Distance from A051650(n) to nearest prime.

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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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A261525 a(n) = smallest m such that A031131(m) = 2*n.

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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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