A051699 -id:A051699 - cp's OEIS frontend

A077019 a(n) is the smallest number for which the prime distance A051699 is equal to n.

2, 1, 0, 26, 93, 118, 119, 120, 531, 532, 897, 1140, 1339, 1340, 1341, 1342, 1343, 1344, 9569, 15702, 15703, 15704, 15705, 19632, 19633, 19634, 19635, 31424, 31425, 31426, 31427, 31428, 31429, 31430, 31431, 31432, 31433, 155958, 155959, 155960, 155961, 155962, 155963, 155964

Offset: 0

Eric W. Weisstein, Oct 17 2002

nonn

David A. Corneth, Table of n, a(n) for n = 0..775 (using b-file from A002386)
Eric Weisstein's World of Mathematics, Prime Distance

Cf. A002386, A051699.

d(n) = if(n<1, 2*(n==0), min(nextprime(n)-n, n-precprime(n))); \\ A051699
a(n) = my(k=0); while (d(k) != n, k++); k; \\ Michel Marcus, Aug 21 2019

More terms from Michel Marcus, Aug 21 2019

A110976 Sequence of numerators associated with the continued fraction based on the sequence d(n)= distance of n from closest prime ( A051699).

Original entry on oeis.org

2, 3, 2, 3, 5, 3, 8, 3, 11, 25, 36, 25, 61, 25, 86, 197, 283, 197, 480, 197, 677, 1551, 2228, 1551, 3779, 9109, 31106, 71321, 102427, 71321, 173748, 71321, 245069, 561459, 1929446, 4420351, 6349797, 4420351, 10770148, 25960647, 36730795, 25960647

Offset: 0

Giorgio Balzarotti and Paolo P. Lava, Sep 28 2005

The value of the continued fraction (for n to infinity) is 2.77459638163600405370875399896...; A(n) = A(n+2) if d(n) =2 and d(n+2) = 0

			if n = 2, A(n) = A(2) = 3 because A(0) = 2, A(1) = 1 * A(0) + 1 = 3, as the distances of n from closest prime are 2, 1, 0, 0, 1 ...

G. Balzarotti and P. P. Lava, Le sequenze di numeri interi, Hoepli, 2008, p. 110.

Cf. A051699, A109139, A109140, A110977.

A[0]:=d[0]; A[1]:=d[1]*A[0]+1; B[0]:=1; B[1]:=d[1]*B[0]; for n from 2 by 1 to N do A[n]:=d[n]*A[n-1]+A[n-2]; B[n]:=d[n]*B[n-1]+B[n-2]; od;

See program

A110977 Sequence of denominators associated with the continued fraction based on the sequence d(n)= distance of n from closest prime ( A051699).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 1, 4, 9, 13, 9, 22, 9, 31, 71, 102, 71, 173, 71, 244, 559, 803, 559, 1362, 3283, 11211, 25705, 36916, 25705, 62621, 25705, 88326, 202357, 695397, 1593151, 2288548, 1593151, 3881699, 9356549, 13238248, 9356549, 22594797, 9356549

Offset: 0

Giorgio Balzarotti and Paolo P. Lava, Sep 28 2005

			if n = 2, B(n) = B(2) = 1 because B(0) = 1, B(1) = 1 * B(0) = 1 as the distances of n from closest prime are 2, 1, 0, 0, 1 ...

G. Balzarotti and P. P. Lava, Le sequenze di numeri interi, Hoepli, 2008, p. 110.

Cf. A051699, A109139, A109140, A110976.

A[0]:=d[0]; A[1]:=d[1]*A[0]+1; B[0]:=1; B[1]:=d[1]*B[0]; for n from 2 by 1 to N do A[n]:=d[n]*A[n-1]+A[n-2]; B[n]:=d[n]*B[n-1]+B[n-2]; od;

See program.

A359734 Lexicographically earliest sequence of distinct nonnegative integers such that the sequence A051699(a(n)) (distance from the nearest prime) has the same sequence of digits.

Original entry on oeis.org

1, 10, 2, 0, 3, 26, 9, 119, 532, 4, 6, 896, 118, 34, 15, 93, 121, 531, 898, 205, 8, 12, 533, 50, 117, 14, 122, 1078, 56, 16, 21, 18, 144, 64, 20, 895, 1138, 899, 25, 5, 186, 1077, 22, 27, 204, 76, 86, 206, 7, 24, 28, 120, 30, 123, 32, 33, 35, 36, 11, 300

Offset: 0

M. F. Hasler and Eric Angelini, Jan 12 2023

In the definition, "has the same digits" means that the concatenation of the terms yields the same string of digits, for the sequence a(.) and the sequence A051699(a(.)).

Conjectured to be a permutation of the nonnegative integers. The inverse permutation would start (3, 0, 2, 4, 9, 39, 10, 48, 20, 6, 1, 58, 21, 75, 25, 14, ...).

			Below, row "p" lists the closest prime to a(n) and row "d" the absolute difference |a(n)-p|. We have the same sequence of digits in rows a (this sequence) and d:
  n :  0   1   2   3   4   5   6   7   8   9  10  11  12  13  14 ...
  a :  1  10   2   0   3  26   9  119 532  4   6 896 118  34  15 ...
  p :  2  11   2   2   3  23   7  113 523  3   5 887 113  31  13 ...
  d :  1   1   0   2   0   3   2   6   9   1   1   9   5   3   2 ...

Eric Angelini, Digit-spines, personal blog "Cinquante signes" on blogspot.com, Jan. 11, 2023.
Eric Angelini, Digit-spines, personal blog "Cinquante signes" on blogspot.com, Jan. 11, 2023. [Cached copy]

Cf. A051699 (distance from the nearest prime), A000040 (the primes).

Cf. A359736, A359737 (similar for squares and Fibonacci numbers).

spine(f, N=20, S=[], d=[], md = n -> if(n, digits(n), [0])) = { vector(N, n, my(m, j=1); for(k=0, oo, setsearch(S, k) && next; while( f(j) < k, j++); m = md(min(m = f(j) - k, iferr(k - f(j-1), E, m))); if(m == concat(d, md(k))[1..#m], d = concat(d, md(k))[#m+1 .. -1]; m=k; break)); S = setunion(S, [m]); m)}
spine(prime, 200) \\ 200 terms of this sequence

A072926 a(n) = Sum_{k=1..n} A051699(k).

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 3, 4, 6, 7, 7, 8, 8, 9, 11, 12, 12, 13, 13, 14, 16, 17, 17, 18, 20, 23, 25, 26, 26, 27, 27, 28, 30, 33, 35, 36, 36, 37, 39, 40, 40, 41, 41, 42, 44, 45, 45, 46, 48, 51, 53, 54, 54, 55, 57, 60, 62, 63, 63, 64, 64, 65, 67, 70, 72, 73, 73, 74, 76, 77, 77, 78, 78, 79

Offset: 1

Benoit Cloitre, Aug 11 2002

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A051699.

f[n_] := If[PrimeQ[n], 0, Min[NextPrime[n] - n, n - NextPrime[n, -1]]]; Accumulate[Table[f[n], {n, 1, 100}]] (* Amiram Eldar, May 05 2022 *)

a(n)=sum(k=1,n,vecmin(vector(k,i,abs(k-prime(i)))))

from sympy import nextprime; p = a = 1
while p < 71:
    q = nextprime(p); h = (q - p)//2
    for i in range(q-p): a += h - abs(h-i); print(a, end = ', ')
    p = q # Ya-Ping Lu, Apr 06 2025

Conjecture: a(n) is asymptotic to C*n*log(n) with C = 0.29... .
From Ya-Ping Lu, Apr 06 2025: (Start)
C = lim_{n->oo} a(n)/(n*log(n)) = 0.44 approximately.
a(prime(m)) = 1 + Sum_(i=3..m) (1/4)*(prime(i)-prime(i-1))^2. (End)

A110917 Conversion to a regular-simple continued-fraction approximation of the limit value (C0=2.7745963816360040537087...) of the continued fraction (numerator = A110976 and denominator = A110977) based on the sequence of the distances of n from closest primes (A051699).

Original entry on oeis.org

2, 1, 3, 2, 3, 2, 3, 2, 2, 2, 3, 2, 3, 2, 3, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 3, 2, 3, 2, 3, 2, 2, 2, 3, 2, 3, 2, 2, 2, 2, 2, 3, 2, 2, 2, 3, 4, 3, 2, 2, 2, 3, 2, 3, 2, 2, 2, 3, 4, 5, 6, 7, 6, 5, 4, 3, 2, 2, 2, 2, 2, 3, 2, 3, 2, 3, 4, 5, 4, 3, 2, 3, 2, 3, 2, 2, 2, 3, 2, 2, 2, 2, 2, 3, 2, 2, 2, 3, 2, 3, 2, 3, 4, 5

Offset: 1

Giorgio Balzarotti and Paolo P. Lava, Oct 04 2005

With the exception of n = 3, it should be abs(a(n)-a(n-1)) = < 1 for all n. Hill-mountain-like plot, with land = 2.

			C0 = a(1) +1/( a(2) +1/( a(3) +1/( a(4) +1/( a(5) +...=2+1/(1+1/(3+1/(2+1/(3+...

G. Balzarotti and P. P. Lava, Le sequenze di numeri interi, Hoepli, 2008, p. 110.

Cf. A051699, A110976, A110977.

cd:=proc(N) # d[n]distance of n from closest prime A[0]:=d[0]; A[1]:=d[1]*A[0]+1; B[0]:=1; B[1]:=d[1]*B[0]; for n from 2 by 1 to N do A[n]:=d[n]*A[n-1]+A[n-2]; B[n]:=d[n]*B[n-1]+B[n-2]; od; R:=A[N]/B[N]; convert(R,confrac); end:

see program

A308633 Continued fraction for the decimal expansion of the concatenation of the terms of A051699 (distance from n to closest prime).

Original entry on oeis.org

0, 4, 1, 3, 5, 9, 1, 2, 2, 4, 7, 1, 246, 1, 2, 2, 1, 116363868, 3, 1, 1, 1, 3, 4, 282, 1, 1, 1, 2, 1, 8, 2, 1, 1, 1, 1, 7, 10, 7, 1, 2, 1, 6, 2, 1, 2, 7, 2, 11, 1, 3, 1, 4, 1, 4, 1, 3, 5, 9, 1, 1, 1, 3, 3, 1, 3, 2, 1, 5, 3, 3, 1, 32, 1, 1, 15, 3, 1, 1, 11, 9, 1

Offset: 0

Paolo P. Lava, Jun 17 2019

Continued fraction for .2100101012101012101012101232101012321... (see A051699).

Very high value for a(17) = 116363868. This should imply that using the first 16 terms we have a good rational approximation of this decimal expansion: 131256182/624999375 is ok up to the 25th decimal digit.

Cf. A030168, A051699.

Digits:=200: with(numtheory): P:=proc(q) local a,b,n; a:=21;
for n from 2 to q do if isprime(n) then a:=10*a; else
b:=min(nextprime(n)-n,n-prevprime(n)); a:=a*10^length(b)+b; fi; od;
op(convert(evalf(a/10^length(a)),confrac,100)); end: P(200);

A080733 Smallest distance from n to a squarefree number.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 2, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 2, 1, 0, 0, 0, 1, 0

Offset: 1

Benoit Cloitre, Mar 08 2003

nonn

a(n) = min (abs(n-k) : where k runs through the squarefree numbers ).
The sequence is unbounded.
The first 0 occurs at 1, the first 1 at 4, the first 2 at 49, the first 3 at 846. - Antti Karttunen, Sep 22 2017

			For n = 3, 3 itself is a squarefree number, thus a(3) = 0.
For n = 48, 48 = 2^4 * 3 is not squarefree, 49 = 7^2 is not squarefree, but 47 is, thus a(48) = abs(48-47) = 1.
For n = 49, neither 49 = 7^2, nor 48 = 2^4 * 3 nor 50 = 2^2 * 5 is squarefree, while both 47 and 51 are, thus a(49) = abs(49-47) = abs(49-51) = 2.

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

Cf. A051699.

Cf. A005117, A081221.

nn=110;With[{sqfr=Select[Range[nn+10],SquareFreeQ]},Flatten[Table[ Union[ Abs[ Nearest[ sqfr,n]-n]],{n,nn}]]] (* Harvey P. Dale, Jun 01 2012 *)

A080733(n) = { my(k=0); while((!issquarefree(n+k))&&(!issquarefree(n-k)),k++); k; }; \\ Antti Karttunen, Sep 22 2017

a(A005117(n)) = 0.

Examples added by Antti Karttunen, Sep 22 2017

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

A080732 Smallest distance from n to a prime power (as defined in A246655).

Original entry on oeis.org

1, 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

Offset: 1

Benoit Cloitre, Mar 08 2003

nonn

a(n)=min (abs(n-k) : where k runs through the prime powers)

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A051699, A246547, A301295.

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). When you refer to "prime powers", be sure to specify which of these you mean. - N. J. A. Sloane, Mar 24 2018

nn = 100; pp = Select[Range[2, Prime[1 + PrimePi[nn]]], Length[FactorInteger[#]] == 1 &]; Table[Min[Abs[n - pp]], {n, nn}] (* T. D. Noe, Mar 14 2012 *)

cp's OEIS Frontend

A077019 a(n) is the smallest number for which the prime distance A051699 is equal to n.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Extensions

A110976 Sequence of numerators associated with the continued fraction based on the sequence d(n)= distance of n from closest prime ( A051699).

Views

Author

Keywords

Comments

Examples

References

Crossrefs

Programs

Maple

Formula

A110977 Sequence of denominators associated with the continued fraction based on the sequence d(n)= distance of n from closest prime ( A051699).

Views

Author

Keywords

Examples

References

Crossrefs

Programs

Maple

Formula

A359734 Lexicographically earliest sequence of distinct nonnegative integers such that the sequence A051699(a(n)) (distance from the nearest prime) has the same sequence of digits.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A072926 a(n) = Sum_{k=1..n} A051699(k).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

A110917 Conversion to a regular-simple continued-fraction approximation of the limit value (C0=2.7745963816360040537087...) of the continued fraction (numerator = A110976 and denominator = A110977) based on the sequence of the distances of n from closest primes (A051699).

Views

Author

Keywords

Comments

Examples

References

Crossrefs

Programs

Maple

Formula

A308633 Continued fraction for the decimal expansion of the concatenation of the terms of A051699 (distance from n to closest prime).

Views

Author

Keywords

Comments

Crossrefs

Programs

Maple

A080733 Smallest distance from n to a squarefree number.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs