A051700 -id:A051700 - 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

A051652 Smallest number at distance n from nearest prime.

Original entry on oeis.org

2, 1, 0, 26, 23, 118, 53, 120, 409, 532, 293, 1140, 211, 1340, 1341, 1342, 1343, 1344, 2179, 15702, 3967, 15704, 15705, 19632, 16033, 19634, 19635, 31424, 31425, 31426, 24281, 31428, 31429, 31430, 31431, 31432, 31433, 155958, 155959, 155960, 38501

Offset: 0

N. J. A. Sloane

Michael S. Branicky, Table of n, a(n) for n = 0..228 (terms 0..91 from _R. J. Mathar_)
Michael S. Branicky, Python program

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

A051700 := proc(m) option remember ; if m <= 2 then op(m+1,[2,1,1]) ; else min(nextprime(m)-m,m-prevprime(m)) ; fi ; end:
A051652 := proc(n) local m ; if n = 0 then RETURN(2); else for m from 0 do if A051700(m) = n then RETURN(m) ; fi ; od: fi ; end:
for n from 0 to 79 do printf("%d %d\n",n,A051652(n)); od: # R. J. Mathar, Jul 22 2009

A051700[n_] := A051700[n] = Min[ NextPrime[n] - n, n - NextPrime[n, -1]]; a[n_] := For[m = 0, True, m++, If[A051700[m] == n, Return[m]]]; a[0] = 2; Table[ a[n], {n, 0, 40}] (* Jean-François Alcover, Dec 19 2011, after R. J. Mathar *)
Join[{2,1,0},Drop[Flatten[Table[Position[Table[Min[NextPrime[n]-n, n-NextPrime[ n,-1]],{n,200000}],?(#==i&),{1},1],{i,40}]],2]] (* _Harvey P. Dale, Mar 16 2015 *)

# 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) != n: m += 1
  return m
print([a(n) for n in range(26)]) # Michael S. Branicky, Feb 27 2021

More terms from James Sellers, Dec 07 1999

A163492 Numbers such that the two adjacent integers are a perfect square and a prime.

Original entry on oeis.org

1, 2, 3, 8, 10, 24, 48, 80, 82, 168, 224, 226, 360, 440, 442, 728, 840, 1088, 1090, 1224, 1368, 1522, 1848, 2026, 2208, 2400, 3024, 3250, 3720, 3968, 4760, 5040, 5624, 5928, 6562, 7920, 8648, 9802, 10608, 11026, 11448, 12322, 13688, 13690, 14160, 14640

Offset: 1

Gaurav Kumar, Jul 29 2009

nonn

Also known as the Beprisque numbers.

			a(1) = 2 since 2 lies between 1(square) and 3(prime).
a(2) = 3 since 3 lies between 2(prime) and 4(square).

T. D. Noe, Table of n, a(n) for n = 1..1000
Planet Math, Beprisque numbers

Cf. A051700, A066635.

nn = 100; Sort[Select[Range[0, nn], PrimeQ[#^2 + 2] &]^2 + 1, Select[Range[nn], PrimeQ[#^2 - 2] &]^2 - 1] (* T. D. Noe, Aug 29 2012 *)
Select[Range[15000],AnyTrue[#+{1,-1},PrimeQ]&&AnyTrue[{Sqrt[#-1],Sqrt[ #+1]},IntegerQ]&] (* Harvey P. Dale, Feb 06 2023 *)

Definition clarified by R. J. Mathar, Aug 08 2009

Number 1 added by WG Zeist, Aug 28 2012

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

A060272 Distance from n^2 to closest prime.

Original entry on oeis.org

1, 1, 2, 1, 2, 1, 2, 3, 2, 1, 6, 5, 2, 1, 2, 1, 4, 7, 2, 1, 2, 3, 6, 1, 6, 1, 2, 3, 2, 7, 6, 3, 2, 3, 2, 1, 2, 3, 2, 1, 12, 5, 2, 3, 2, 3, 2, 5, 2, 3, 8, 3, 6, 1, 2, 1, 2, 3, 10, 7, 2, 3, 2, 3, 4, 1, 4, 3, 2, 3, 2, 5, 4, 1, 2, 3, 2, 5, 6, 3, 2, 5, 6, 1, 4, 3, 4, 3, 2, 1, 6, 3, 2, 1, 4, 5, 4, 3, 2, 7, 8, 5, 2

Offset: 1

Labos Elemer, Mar 23 2001

nonn

			n=1: n^2=1 has next prime 2, so a(1)=1;
n=11: n^2=121 is between primes {113,127} and closer to 127, thus a(11)=6.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A007491, A053001, A058043, A056927, A053000, A051700, A060268-A060269, A060272, A059959, A059790.

seq((s-> min(nextprime(s)-s, `if`(s>2, s-prevprime(s), [][])))(n^2), n=1..256);  # edited by Alois P. Heinz, Jul 16 2017

Table[Function[k, Min[k - #, NextPrime@ # - k] &@ If[n == 1, 0, Prime@ PrimePi@ k]][n^2], {n, 103}] (* Michael De Vlieger, Jul 15 2017 *)
Min[#-NextPrime[#,-1],NextPrime[#]-#]&/@(Range[110]^2) (* Harvey P. Dale, Jun 26 2021 *)

a(n) = if (n==1, nextprime(n^2) - n^2, min(n^2 - precprime(n^2), nextprime(n^2) - n^2)); \\ Michel Marcus, Jul 16 2017

a(n) = abs(A000290(n) - A113425(n)) = abs(A000290(n) - A113426(n)). - Reinhard Zumkeller, Oct 31 2005

A132860 Smallest number at distance 2n from nearest prime (variant 2).

Original entry on oeis.org

2, 0, 93, 119, 531, 897, 1339, 1341, 1343, 9569, 15703, 15705, 19633, 19635, 31425, 31427, 31429, 31431, 31433, 155959, 155961, 155963, 360697, 360699, 360701, 370311, 370313, 370315, 370317, 1349591, 1357261, 1357263, 1357265, 1357267

Offset: 1

R. J. Mathar, Nov 18 2007, Nov 30 2007

nonn

Let f(m) be the distance to the nearest prime as defined in A051699(m). Then a(n) = min { m: f(m)= 2n }. A051728 uses A051700(m) to define the distance.

Note that the requirement f(m)>=2n yields the same sequence as f(m)=2n here. (Reasoning: We are essentially probing for prime gaps of size 4n or larger while increasing m. One cannot get earlier hits by relaxing the requirement from the equal to the larger-or-equal sign, because m triggers as soon as the distance to the start of the gap reaches 2n, with both definitions. This is an inherent consequence of using A051699.)

Cf. A051728, A051699.

A051699 := proc(m) if isprime(m) then 0 ; elif m <= 2 then op(m+1,[2,1]) ; else min(nextprime(m)-m,m-prevprime(m)) ; fi ; end: a := proc(n) local m ; for m from 0 do if A051699(m) = 2 * n then RETURN(m) ; fi ; od: end: seq(a(n),n=0..18);

a(n) = min {m : A051699(m) = 2n}.

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

A163768 Distance of Fibonacci(n) to the closest prime which is not Fibonacci(n) itself.

Original entry on oeis.org

2, 1, 1, 1, 1, 2, 1, 2, 2, 3, 2, 6, 5, 4, 2, 3, 4, 4, 5, 4, 2, 3, 2, 4, 13, 4, 10, 11, 14, 10, 23, 4, 4, 9, 10, 14, 11, 6, 12, 3, 2, 6, 7, 12, 16, 9, 24, 6, 5, 20, 18, 23, 14, 6, 9, 12, 10, 21, 4, 30, 13, 38, 4, 7, 16, 12, 19, 36, 22, 31, 4, 32, 11, 12, 60, 7, 2, 6, 27, 12, 62, 25, 20, 6, 19, 78

Offset: 0

Jonathan Vos Post, Aug 04 2009

The closest prime to F(n) -- next closest if F(n) itself is prime -- for n = 0, 1, 2, 3, 4, ...:

2, 2, 2, 3, 2, 3 or 7, 7, 11, 19 or 23, 31 or 37, 53, 83, 139 or 149, 229, 379, 607 or 613.

			a(0) = 2 because 2 is the closest prime to F(0) = 0, and 2-0 = 2.
a(1) = 1 because 2 is the closest prime to F(1) = 1, and 2-1 = 1.
a(3) = 1 because 3 is the closest prime to F(3) = 2 other than the prime F(3) = 2 itself, and 3-2 = 1.

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A000040, A000045, A001605, A005478, A023186.

A051700 := proc(n) if n < 2 then 2-n; elif n = 2 then 1 ; else min( nextprime(n)-n, n-prevprime(n) ); fi; end:
A000045 := proc(n) combinat[fibonacci](n) ; end:
A163768 := proc(n) A051700(A000045(n)) ; end: seq(A163768(n), n=0..100) ; # R. J. Mathar, Aug 06 2009

g[n_]:=Module[{fn=Fibonacci[n],a,b},a=NextPrime[fn,-1];b=NextPrime[fn];Min[Abs[fn-a],Abs[b-fn]]]; Table[g[i],{i,0,100}] (* Harvey P. Dale, Jan 15 2011 *)

For n not in A001605: a(n) = MIN{|A000045(n) - A000040(i)} = A079677(n).
For n in A001605: a(n) = MIN{k such that k > 0 and |A000045(n) - A000040(i)| = k}.
a(n) = A051700(A000045(n)). - R. J. Mathar, Aug 06 2009

More terms from R. J. Mathar, Aug 06 2009, reformatted Aug 29 2009

A308261 For any integer n, let d(n) be the smallest k > 0 such that at least one of n-k or n+k is a prime number; we build an undirected graph G on top of the prime numbers as follows: two consecutive prime numbers p and q are connected iff at least one of d(p) or d(q) equals q-p; a(n) is the number of terms in the n-th connected component of G (ordered by least element).

Original entry on oeis.org

4, 2, 3, 2, 7, 3, 3, 3, 3, 2, 2, 8, 2, 7, 2, 5, 4, 4, 2, 4, 5, 3, 2, 2, 3, 4, 3, 3, 2, 2, 5, 8, 7, 4, 2, 5, 3, 2, 2, 2, 2, 3, 4, 4, 3, 5, 4, 2, 2, 2, 3, 2, 3, 6, 3, 2, 2, 4, 6, 2, 3, 2, 4, 3, 4, 2, 5, 4, 3, 7, 4, 2, 2, 2, 3, 4, 4, 4, 2, 5, 4, 2, 2, 5, 3, 3, 2

Offset: 1

Rémy Sigrist, Jun 02 2019

nonn

Each connected component of G has at least two elements.

Is the sequence bounded?

			The first terms, alongside the corresponding components, are:
  n  a(n)   n-th component
  -- ----   --------------
   1    4   {2, 3, 5, 7}
   2    2   {11, 13}
   3    3   {17, 19, 23}
   4    2   {29, 31}
   5    7   {37, 41, 43, 47, 53, 59, 61}
   6    3   {67, 71, 73}
   7    3   {79, 83, 89}
   8    3   {97, 101, 103}
   9    3   {107, 109, 113}
  10    2   {127, 131}

Cf. A000040, A051700.

d(p) = for (k=1, oo, if (p-k>0 && isprime(p-k), return (k), isprime(p+k), return (k)))
v=1; p=2; forprime (q=p+1, oo, if (d(p)==q-p || d(q)==q-p, v++, print1 (v", "); if (n++==87, break); v = 1); p=q)

A133490 Smallest number at distance 2n from nearest prime (variant 3).

Original entry on oeis.org

2, 0, 23, 53, 211, 211, 211, 1341, 1343, 2179, 3967, 15705, 16033, 19635, 24281, 24281, 31429, 31431, 31433, 38501, 38501, 58831, 203713, 206699, 206699, 370311, 370313, 370315, 370317, 1272749, 1272749, 1272749, 1357265, 1357267, 2010801

Offset: 0

R. J. Mathar, Nov 30 2007

nonn

Let f(m) be the distance to the nearest prime as defined in A051700(m). Then a(n) = min { m: f(m)>= 2n}. The variants of a different definition of "distance" or demanding equality, f(m)=2n, are in A132860 and A051728.

cp's OEIS Frontend

A051728 Smallest number at distance 2n from nearest prime.

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A051652 Smallest number at distance n from nearest prime.

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A163492 Numbers such that the two adjacent integers are a perfect square and a 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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A060272 Distance from n^2 to closest prime.

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A132860 Smallest number at distance 2n from nearest prime (variant 2).

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

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