A060272 -id:A060272 - cp's OEIS frontend

A172989 Smallest k such that the two numbers n^2 +- k are primes.

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

Offset: 2

Vladimir Joseph Stephan Orlovsky, Feb 07 2010

nonn

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

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

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

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

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}]

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

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

A113426 Greatest prime closest to n^2.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Oct 31 2005

nonn

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

A113425(n) <= a(n).

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

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}]

A113425 Smallest prime closest to n^2.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Oct 31 2005

nonn

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

a(n) <= A113426(n).

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

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

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 *)

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

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A172989 Smallest k such that the two numbers n^2 +- k are primes.

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A113426 Greatest prime closest to n^2.

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A113425 Smallest prime closest to n^2.

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

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