A060272 -id:A060272

Search: a060272 -id:a060272

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A172989

Smallest k such that the two numbers n^2 +- k are primes.

+10
9

1, 2, 3, 6, 5, 12, 3, 2, 3, 18, 5, 12, 3, 2, 15, 18, 7, 12, 21, 2, 63, 42, 55, 6, 15, 10, 27, 12, 19, 78, 15, 2, 93, 12, 5, 78, 15, 10, 21, 12, 23, 18, 57, 14, 27, 30, 7, 120, 117, 8, 15, 42, 37, 24, 27, 58, 93, 18, 7, 12, 75, 38, 3, 6, 7, 132, 27, 28, 69, 18, 5, 102, 27, 34, 75, 78, 5

(list; graph; refs; listen; history; text; internal format)

OFFSET

2,2

LINKS

Hugo Pfoertner, Table of n, a(n) for n = 2..10000

FORMULA

a(n) = A082467(n^2). - Ivan N. Ianakiev, Jul 28 2019

EXAMPLE

2^2 +- 1 are both prime, 3^2 +- 2 are both prime, 4^2 +- 3 are both prime, 5^2 +- 6 are both prime, ...

MATHEMATICA

f[n_]:=Block[{k}, If[OddQ[n], k=2, k=1]; While[ !PrimeQ[n-k]||!PrimeQ[n+k], k+=2]; k]; Table[f[n^2], {n, 2, 40}]

PROG

(PARI) a(n) = my(k=1); while(!isprime(n^2+k) || !isprime(n^2-k), k++); k; \\ Michel Marcus, May 20 2018

(Magma) sol:=[]; for m in [2..80] do for k in [1..200] do if IsPrime(m^2-k) and IsPrime(m^2+k) then sol[m-1]:=k; break; end if; end for; end for; sol; // Marius A. Burtea, Jul 28 2019

CROSSREFS

Cf. A060272 (at least one prime), A082467 (supersequence).

KEYWORD

nonn

AUTHOR

Vladimir Joseph Stephan Orlovsky, Feb 07 2010

STATUS

approved

A113426

Greatest prime closest to n^2.

+10
4

2, 5, 11, 17, 23, 37, 47, 67, 83, 101, 127, 149, 167, 197, 227, 257, 293, 331, 359, 401, 443, 487, 523, 577, 631, 677, 727, 787, 839, 907, 967, 1021, 1091, 1153, 1223, 1297, 1367, 1447, 1523, 1601, 1693, 1759, 1847, 1933, 2027, 2113, 2207, 2309, 2399, 2503

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

A060272(n) = abs(A000290(n) - a(n));

A113425(n) <= a(n).

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..1000

MATHEMATICA

f[n_]:=Module[{n2=n^2, np1, np2}, np1=NextPrime[n2, -1]; np2=NextPrime[n2]; If[(n2-np1)<(np2-n2), np1, np2]]

Table[f[i], {i, 50}]

KEYWORD

nonn

AUTHOR

Reinhard Zumkeller, Oct 31 2005

STATUS

approved

A113425

Smallest prime closest to n^2.

+10
3

2, 3, 7, 17, 23, 37, 47, 61, 79, 101, 127, 139, 167, 197, 223, 257, 293, 317, 359, 401, 439, 487, 523, 577, 619, 677, 727, 787, 839, 907, 967, 1021, 1087, 1153, 1223, 1297, 1367, 1447, 1523, 1601, 1669, 1759, 1847, 1933, 2027, 2113, 2207, 2309, 2399, 2503

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

A060272(n) = abs(A000290(n) - a(n));

a(n) <= A113426(n).

LINKS

Robert Israel, Table of n, a(n) for n = 1..10000

MAPLE

f:= proc(n) local k, d;

for k from 1 do

for d in [-1, 1] do

if isprime(n^2 + k*d) then return n^2 + k*d fi

od od

end proc:

map(f, [$1..100]); # Robert Israel, Mar 10 2017

MATHEMATICA

sp[n_]:=Module[{n2=n^2 , npu, npd}, npu=NextPrime[n2]; npd=NextPrime[n2, -1]; If[n2-npd<=npu-n2, npd, npu]]; sp/@Range[50] (* Harvey P. Dale, Feb 05 2011 *)

KEYWORD

nonn

AUTHOR

Reinhard Zumkeller, Oct 31 2005

STATUS

approved

A079666

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

+10
1

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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)

LINKS

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

MAPLE

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

PROG

(PARI) 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)

CROSSREFS

Cf. A051699, A060272.

KEYWORD

nonn

AUTHOR

Benoit Cloitre, Jan 26 2003

EXTENSIONS

More terms from Robert Israel, Jan 03 2017

STATUS

approved

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