A051699 -id:A051699 - cp's OEIS frontend

A296239 a(n) = distance from n to nearest Fibonacci number.

0, 0, 0, 0, 1, 0, 1, 1, 0, 1, 2, 2, 1, 0, 1, 2, 3, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 16, 15, 14, 13, 12, 11, 10, 9

Offset: 0

Rémy Sigrist, Dec 09 2017

The Fibonacci numbers correspond to sequence A000045.
This sequence is analogous to:
- A051699 (distance to nearest prime),
- A053188 (distance to nearest square),
- A053646 (distance to nearest power of 2),
- A053615 (distance to nearest oblong number),
- A053616 (distance to nearest triangular number),
- A061670 (distance to nearest power),
- A074989 (distance to nearest cube),
- A081134 (distance to nearest power of 3),
The local maxima of the sequence correspond to positive terms of A004695.
a(n) = 0 iff n = A000045(k) for some k >= 0.
a(n) = 1 iff n = A061489(k) for some k > 4.
For any n >= 0, abs(a(n+1) - a(n)) <= 1.
For any n > 0, a(n) < n, and a^k(n) = 0 for some k > 0 (where a^k denotes the k-th iterate of a); k equals A105446(n) for n = 1..80 (and possibly more values).
a(n) > max(a(n-1), a(n+1)) iff n = A001076(k) for some k > 1.

			For n = 42:
- A000045(9) = 34 <= 42 <= 55 = A000045(10),
- a(42) = min(42 - 34, 55 - 42) = min(8, 13) = 8.

Cf. A000045, A001076, A004695, A051699, A053188, A053615, A053616, A053646, A061489, A061670, A074989, A081134, A105446.

fibPi[n_] := 1 + Floor[ Log[ GoldenRatio, 1 + n*Sqrt@5]]; f[n_] := Block[{m = fibPi@ n}, Min[n - Fibonacci[m -1], Fibonacci[m] - n]]; Array[f, 81, 0] (* Robert G. Wilson v, Dec 11 2017 *)
With[{nn=80,fibs=Fibonacci[Range[0,20]]},Table[Abs[n-Nearest[fibs,n]][[1]],{n,0,nn}]] (* Harvey P. Dale, Jul 02 2022 *)

a(n) = for (i=1, oo, if (n<=fibonacci(i), return (min(n-fibonacci(i-1), fibonacci(i)-n))))

a(n) = abs(n - Fibonacci(floor(log(sqrt(20)*n)/log((1 + sqrt(5))/2)-1))). - Jon E. Schoenfield, Dec 14 2017

A079677 Distance from F(n) to closest prime, where F(n) is the n-th Fibonacci number.

Original entry on oeis.org

2, 1, 1, 0, 0, 0, 1, 0, 2, 3, 2, 0, 5, 0, 2, 3, 4, 0, 5, 4, 2, 3, 2, 0, 13, 4, 10, 11, 14, 0, 23, 4, 4, 9, 10, 14, 11, 6, 12, 3, 2, 6, 7, 0, 16, 9, 24, 0, 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, 0, 19, 78, 6

Offset: 0

Benoit Cloitre, Jan 26 2003

nonn

T. D. Noe, Table of n, a(n) for n = 0..1000

Cf. A051699, A000045.

Table[f = Fibonacci[n]; If[PrimeQ[f], 0, Min[f - NextPrime[f, -1], NextPrime[f] - f]], {n, 0, 100}] (* _T. D. Noe, May 02 2012 *)

a(s)=min(abs(precprime(fibonacci(s))-fibonacci(s)),abs(nextprime(fibonacci(s))-fibonacci(s)))

A132470 Smallest number at distance exactly 3n from nearest prime.

Original entry on oeis.org

2, 26, 119, 532, 1339, 1342, 9569, 15704, 19633, 31424, 31427, 31430, 31433, 155960, 155963, 360698, 360701, 370312, 370315, 492170, 1357261, 1357264, 1357267, 2010802, 2010805, 4652428, 17051785, 17051788, 17051791, 17051794, 17051797, 20831416, 20831419, 20831422

Offset: 0

Jonathan Vos Post, Sep 03 2007

nonn

Let f(m)= A051699(m) = exact distance from m to its closest prime (including m itself). Then a(n) = min { m : f(m) = 3n}. - R. J. Mathar, Nov 18 2007

This sequence can be derived from the record prime gap sequences A002386 and A005250. In particular, for n > 0, a(n) = A002386(k) + 3*n where k is the least index such that A005250(k) >= 3*n. - Andrew Howroyd, Jan 04 2020

			a(3)=532 where 532+3*3 is prime and all numbers below 532 have a distance smaller or larger than 3n=9 to their nearest primes and there is no prime within a distance of 8 to 532.

Andrew Howroyd, Table of n, a(n) for n = 0..258

Cf. A002386, A005250, A051699, A051728.

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: A132470 := proc(n) local m ; if n = 0 then RETURN(2); else for m from 0 do if A051699(m) = 3 * n then RETURN(m) ; fi ; od: fi ; end: seq(A132470(n),n=0..18) ; # R. J. Mathar, Nov 18 2007

terms = 34;
gaps = Cases[Import["https://oeis.org/A002386/b002386.txt", "Table"], {, }][[;; terms, 2]];
w[n_] := (NextPrime[gaps[[n]] + 1] - gaps[[n]])/6 // Floor;
k = 1; a[0] = 2;
For[n = 1, n <= terms, n++, While[w[k] < n, k++]; a[n] = gaps[[k]] + 3n];
a /@ Range[0, terms-1] (* Jean-François Alcover, Apr 09 2020, after Andrew Howroyd *)

\\ here R(gaps) wants prefix of A002386 as vector.
aA002386(lim)={my(L=List(),q=2,g=0); forprime(p=3, lim, if(p-q>g, listput(L,q); g=p-q); q=p); Vec(L)}
R(gaps)={my(w=vector(#gaps, n, nextprime(gaps[n]+1) - gaps[n])\6, r=vector(w[#w]+1), k=1); r[1]=2; for(n=1, w[#w], while(w[k]A002386(10^7))} \\ Andrew Howroyd, Jan 04 2020

a(n) = min {m : A051699(m) = 3n}. - R. J. Mathar, Nov 18 2007

Corrected by Dean Hickerson, Sep 05 2007
Both this sequence and A051728 should be checked. There are two possibilities for confusion in each case. In defining f(m), does one allow or exclude m itself, in case m is a prime? In defining a(n), does one require (here) that f(m) = 3n or only that >= 3n, or (in A051728) that f(m) = 2n or only >= 2n? Probably there should be several sequences, to include all the possibilities in each case. - N. J. A. Sloane, Nov 18 2007. Added Nov 20 2007: R. J. Mathar has now clarified the definition of the present sequence.
Corrected and extended by R. J. Mathar, Nov 18 2007
Terms a(19) and beyond from Andrew Howroyd, Jan 04 2020

A163497 Numbers n with following property: let c = nearest cube to n that is different from n and let p = nearest prime to n that is different from n. Then |n-c| = |n-p|.

Original entry on oeis.org

2, 25, 28, 119, 126, 340, 345, 728, 731, 1329, 1346, 2188, 2200, 3374, 3382, 4911, 4916, 6858, 6861, 9259, 9269, 12165, 12182, 15622, 15627, 19682, 19685, 24384, 24390, 29790, 29797, 35935, 35944, 42869, 42887, 50652, 50662, 59300, 59326

Offset: 1

Gaurav Kumar, Jul 29 2009

nonn

With the exception of 2 those k where A051699(k) = A074989(k) (same distance to nearest prime and to nearest cube). - R. J. Mathar, Aug 08 2009

			a(1) = 2 since 2 lies between 1 (cube) and 3 (prime);
a(2) = 28 since 28 lies between 27 (cube) and 29 (prime).

Cf. A154840.

A163497 := proc(n) option remember ; local a; if n = 1 then 2; else for a from procname(n-1)+1 do if A051699(a) = A074989(a) then return a; end if; end do ; end if; end proc: # R. J. Mathar, Nov 01 2009

Edited by Zak Seidov, Aug 01 2009

Further edited by N. J. A. Sloane, Oct 31 2009

A301295 Smallest distance from n to a prime power (as defined in A000961).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 2, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 2, 2, 1, 0, 1, 2, 1, 0, 1, 0, 1, 2, 1, 0, 1, 0, 1, 2, 1, 0, 1, 2, 3, 2, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 0, 1, 2, 3, 2, 1, 0, 1, 0, 1, 0, 1, 2, 3, 2, 1, 0, 1, 2, 3, 4, 3, 2, 1, 0, 1, 2, 1, 0, 1, 0, 1, 2

Offset: 1

N. J. A. Sloane, Mar 24 2018

Antti Karttunen, Table of n, a(n) for n = 1..65537

There are four different sequences which may legitimately be called "prime powers": A000961 (p^k, k >= 0), A246655 (p^k, k >= 1), A246547 (p^k, k >= 2), A025475 (p^k, k=0 and k >= 2).
Identical to A080732 except here a(1)=0.
Cf. also A051699, A175851.

A301295(n) = if(1==n,0,my(k=0);while(!isprimepower(n+k) && !isprimepower(n-k), k++); (k)); \\ Antti Karttunen, Sep 25 2018

More terms from Antti Karttunen, Sep 25 2018

A366092 Distance from the sum of the first n primes to the nearest prime.

Original entry on oeis.org

2, 0, 0, 1, 0, 1, 0, 1, 2, 1, 2, 3, 0, 1, 0, 3, 2, 1, 2, 1, 2, 3, 4, 3, 4, 1, 2, 5, 2, 1, 4, 1, 4, 1, 2, 3, 4, 5, 2, 3, 2, 5, 2, 1, 2, 1, 2, 3, 2, 1, 2, 1, 2, 3, 2, 1, 2, 1, 10, 1, 0, 11, 2, 1, 0, 3, 2, 3, 2, 7, 2, 1, 2, 3, 4, 3, 2, 3, 4, 5, 2, 5, 4, 3, 10, 3

Offset: 0

Paolo Xausa, Sep 29 2023

Positions of zeros are given by A013916.

Positions of records are given by A366093.

			a(3) = 1 because the sum of the first 3 primes is 2 + 3 + 5 = 10, the nearest prime is 11 and 11 - 10 = 1.

Paolo Xausa, Table of n, a(n) for n = 0..10000

Cf. A000040, A007504, A013916, A051699, A366093, A366094.

Cf. also A351830, A354330.

pDist[n_]:=If[PrimeQ[n],0,Min[NextPrime[n]-n,n-NextPrime[n,-1]]];
A366092list[nmax_]:=Map[pDist,Prepend[Accumulate[Prime[Range[nmax]]],0]];
A366092list[100]

from sympy import prime, nextprime, prevprime
def A366092(n): return min((m:=sum(prime(i) for i in range(1,n+1)))-prevprime(m+1),nextprime(m)-m) if n else 2 # Chai Wah Wu, Oct 03 2023

a(n) = A051699(A007504(n)).

a(n) = abs(A007504(n) - A366094(n)).

A336364 Rectangular array by antidiagonals: row n shows the positive integers whose distance to the nearest prime is n.

Original entry on oeis.org

2, 3, 1, 5, 4, 9, 7, 6, 15, 26, 11, 8, 21, 34, 93, 13, 10, 25, 50, 117, 118, 17, 12, 27, 56, 123, 122, 119, 19, 14, 33, 64, 143, 144, 121, 120, 23, 16, 35, 76, 145, 186, 205, 300, 531, 29, 18, 39, 86, 185, 204, 217, 324, 533, 532, 31, 20, 45, 92, 187, 206

Offset: 1

Clark Kimberling, Jul 19 2020

Row 1: the primes, A000040. Every positive integer occurs exactly once, so that as a sequence, this is a permutation of the positive integers.

			Corner:
   2   3   5   7  11   13   17   19   23   29   31   37
   1   4   6   8  10   12   14   16   18   20   22   24
   9  15  21  25  27   33   35   39   45   49   51   55
  26  34  50  56  64   76   86   92   94  116  124  134
  93 117 123 143 145  185  187  203  207  215  219  245

Sean A. Irvine, Java program (github).
Index entries for sequences that are permutations of the natural numbers

Cf. A000040, A051699, A336365.

a[?PrimeQ] = 0; a[n] := Min[NextPrime[n] - n, n - NextPrime[n, -1]];
t = Table[a[n], {n, 1, 2000}]; (* A051699 *)
r[n_] := Flatten[Position[t, n]]; u[n_, k_] := r[n][[k]];
TableForm[Table[u[n, k], {n, 0, 15}, {k, 1, Length[r[n]]}]] (* A337364, array *)
Table[u[n - k, k], {n, 0, 15}, {k, n, 1, -1}] // Flatten    (* A337364, sequence *)

A336365 Rectangular array by antidiagonals: row n shows the nonnegative integers whose distance to the nearest prime is n.

Original entry on oeis.org

2, 3, 1, 5, 4, 0, 7, 6, 9, 26, 11, 8, 15, 34, 93, 13, 10, 21, 50, 117, 118, 17, 12, 25, 56, 123, 122, 119, 19, 14, 27, 64, 143, 144, 121, 120, 23, 16, 33, 76, 145, 186, 205, 300, 531, 29, 18, 35, 86, 185, 204, 217, 324, 533, 532, 31, 20, 39, 92, 187, 206

Offset: 1

Clark Kimberling, Jul 19 2020

Row 1: the primes, A000040.

Every nonnegative integer occurs exactly once, so that as a sequence, this is a permutation of the nonnegative integers.

			 Corner:
   2    3    5    7   11   13   17   19   23   29   31   37
   1    4    6    8   10   12   14   16   18   20   22   24
   0    9   15   21   24   27   33   35   39   45   49   51
  26   34   50   56   64   76   86   92   94  116  124  134
  93  117  123  143  145  185  187  203  207  215  219  245

Cf. A000040, A051699, A336364.

a[?PrimeQ] = 0; a[n] := Min[NextPrime[n] - n, n - NextPrime[n, -1]];
t = Table[a[n], {n, 0, 2000}]; (*  A051699 *)
r[n_] := -1 + Flatten[Position[t, n]]; u[n_, k_] := r[n][[k]];
TableForm[Table[u[n, k], {n, 0, 15}, {k, 1, Length[r[n]]}]] (* A336365, array *)
Table[u[n - k, k], {n, 0, 15}, {k, n, 1, -1}] // Flatten  (* A336365, sequence *)

A079666 Least k such that the distance from k^2 to closest prime = n or zero if no k exists.

Original entry on oeis.org

1, 3, 8, 17, 12, 11, 18, 51, 200, 59, 238, 41, 276, 165, 104, 281, 214, 397, 348, 159, 650, 305, 778, 923, 2242, 1155, 1090, 911, 822, 1871, 1280, 1099, 1516, 3253, 2578, 5849, 3538, 693, 4010, 1937, 1284, 5095, 3212, 2011, 6268, 6331, 2160, 1943, 12470, 13443, 12836, 7405, 25428, 7115, 22596, 10873

Offset: 1

Benoit Cloitre, Jan 26 2003

nonn

From Robert Israel, Jan 03 2017: (Start)
For n > 1, a(n) == n (mod 2) unless it is 0.
a(191) > 3*10^7 if it is not 0. (End)

Robert Israel, Table of n, a(n) for n = 1..190

Cf. A051699, A060272.

N:= 100: # for a(1)..a(N)
R[1]:= 1: count:= 1:
for k from 3 while count < N do
d:= min(nextprime(k^2)-k^2,k^2-prevprime(k^2));
if d <= N and not assigned(R[d]) then R[d]:= k; count:= count+1 fi
od:
seq(R[i],i=1..N); # Robert Israel, Jan 03 2017

a(n)=if(n<0,0,s=1; while(abs(n-min(abs(precprime(s^2)-s^2),abs(nextprime(s^2)-s^2)))>0,s++); s)

More terms from Robert Israel, Jan 03 2017

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}.

cp's OEIS Frontend

A296239 a(n) = distance from n to nearest Fibonacci number.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A079677 Distance from F(n) to closest prime, where F(n) is the n-th Fibonacci number.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

A132470 Smallest number at distance exactly 3n from nearest prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A163497 Numbers n with following property: let c = nearest cube to n that is different from n and let p = nearest prime to n that is different from n. Then |n-c| = |n-p|.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Extensions

A301295 Smallest distance from n to a prime power (as defined in A000961).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Extensions

A366092 Distance from the sum of the first n primes to the nearest prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Python

Formula

A336364 Rectangular array by antidiagonals: row n shows the positive integers whose distance to the nearest prime is n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A336365 Rectangular array by antidiagonals: row n shows the nonnegative integers whose distance to the nearest prime is n.

Views

Author