A075555 -id:A075555 - cp's OEIS frontend

A080883 Distance of n to next square.

1, 3, 2, 1, 5, 4, 3, 2, 1, 7, 6, 5, 4, 3, 2, 1, 9, 8, 7, 6, 5, 4, 3, 2, 1, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 19, 18, 17, 16, 15, 14, 13

Offset: 0

Ralf Stephan, Mar 29 2003

The following sequences all have the same parity: A004737, A006590, A027052, A071028, A071797, A078358, A078446, A080883. - Jeremy Gardiner, Dec 30 2006

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

Cf. A075555.

Cf. A066635, A053188. - R. J. Mathar, Aug 08 2009

List([0..90], n-> Int(1+RootInt(n))^2 -n); # G. C. Greubel, Nov 07 2019

[Floor(1+Sqrt(n))^2 -n: n in [0..90]]; // G. C. Greubel, Nov 07 2019

A080883 := proc(n) (floor(sqrt(n)+1))^2 -n ; end: seq( A080883(n),n=0..40) ; # R. J. Mathar, Aug 08 2009

Table[Floor[1+Sqrt[n]]^2 -n, {n,0,90}] (* G. C. Greubel, Nov 07 2019 *)

a(n) = (sqrtint(n)+1)^2-n; \\ Michel Marcus, May 22 2024

[floor(1+sqrt(n))^2 -n for n in (0..90)] # G. C. Greubel, Nov 07 2019

a(n) = floor( sqrt(n)+1 )^2 - n.

A075556 Smallest prime p not occurring earlier such that p+n is a square, or 0 if no such p exists.

Original entry on oeis.org

3, 2, 13, 5, 11, 19, 29, 17, 7, 71, 53, 37, 23, 67, 181, 0, 47, 31, 557, 61, 43, 59, 41, 97, 0, 199, 73, 197, 167, 139, 113, 89, 163, 191, 109, 0, 107, 83, 157, 401, 103, 79, 101, 317, 151, 179, 149, 241, 0, 239, 349, 173, 271, 307, 269, 233, 619, 383, 137, 229

Offset: 1

Amarnath Murthy, Sep 23 2002

nonn

a(n)=0 or 2*sqrt(n)+1 for square n. Apparently the only cases where it is 2*sqrt(n)+1 are n=1, 4 and 9. - Ralf Stephan, Mar 30 2003, corrected by Robert Israel, Dec 07 2024

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

Cf. A075555, A075557.

for n from 1 to 100 do
  if issqr(n) then
    r:= sqrt(n);
    if isprime(2*r+1) and not assigned(S[2*r+1]) then R[n]:= 2*r+1; S[2*r+1]:= n else R[n] := 0 fi;
  else
    for k from ceil(sqrt(n)) do
      if not assigned(S[k^2-n]) and isprime(k^2-n) then R[n]:= k^2-n; S[k^2-n]:= n; break fi;
    od
  fi;
od:
seq(R[i],i=1..100); # Robert Israel, Dec 06 2024

v=vector(1000000); for(n=1, 100, f=0; forprime(p=2, 1000000, if(!v[p]&&issquare(p+n), f=p; break)); if(f, print1(f", "); v[f]=1, print1("0, ")));

More terms from Ralf Stephan, Mar 30 2003

A105016 Smallest a(n) such that a(n)^2 - n is a positive prime, or 0 if no such a(n) exists.

Original entry on oeis.org

0, 2, 2, 4, 3, 4, 3, 3, 5, 4, 9, 4, 5, 4, 4, 14, 0, 6, 5, 6, 5, 8, 5, 5, 11, 6, 7, 8, 9, 6, 7, 6, 7, 6, 6, 8, 7, 12, 7, 10, 9, 8, 7, 12, 7, 8, 7, 7, 11, 0, 9, 8, 9, 8, 11, 12, 13, 8, 9, 8, 11, 8, 8, 10, 9, 12, 13, 18, 9, 10, 9, 10, 13, 12, 9, 16, 9, 10, 9, 9, 11, 10, 21, 10, 11, 12, 13, 10, 15, 10

Offset: 0

Joshua Zucker, Mar 31 2005

nonn

An old ARML problem asked for the smallest n>0 such that a(n) does not exist.

			a(8) = 5 because 5^2 - 8 = 17 is the smallest square that gives a prime difference.
a(16) = 0 because if x^2 - 16 is prime, then a prime equals (x+4)(x-4), which is impossible.

T. D. Noe, Table of n, a(n) for n = 0..10000

Cf. A075555 for the primes = a(n)^2 - n.

Table[s = Sqrt[n]; If[IntegerQ[s], If[PrimeQ[(s + 1)^2 - n], k = s + 1, k = 0], k = Ceiling[s]; While[! PrimeQ[k^2 - n], k++]]; k, {n, 0, 100}] (* T. D. Noe, Apr 17 2011 *)

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A080883 Distance of n to next square.

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A075556 Smallest prime p not occurring earlier such that p+n is a square, or 0 if no such p exists.

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Maple

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A105016 Smallest a(n) such that a(n)^2 - n is a positive prime, or 0 if no such a(n) exists.

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Mathematica