A069003 -id:A069003 - cp's OEIS frontend

A053000 a(n) = (smallest prime > n^2) - n^2.

2, 1, 1, 2, 1, 4, 1, 4, 3, 2, 1, 6, 5, 4, 1, 2, 1, 4, 7, 6, 1, 2, 3, 12, 1, 6, 1, 4, 3, 12, 7, 6, 7, 2, 7, 4, 1, 4, 3, 2, 1, 12, 13, 12, 13, 2, 13, 4, 5, 10, 3, 8, 3, 10, 1, 12, 1, 2, 7, 10, 7, 6, 3, 20, 3, 4, 1, 4, 13, 22, 3, 10, 5, 4, 1, 14, 3, 10, 5, 6, 21, 2, 9, 10, 1, 4, 15, 4, 9, 6, 1, 6, 3, 14

Offset: 0

N. J. A. Sloane, Feb 21 2000

Suggested by Legendre's conjecture (still open) that there is always a prime between n^2 and (n+1)^2.
Record values are listed in A070317, their indices in A070316. - M. F. Hasler, Mar 23 2013
Conjecture: a(n) <= 1+phi(n) = 1+A000010(n), for n>0. This improves on Oppermann's conjecture, which says a(n) < n. - Jianglin Luo, Sep 22 2023

J. R. Goldman, The Queen of Mathematics, 1998, p. 82.
R. K. Guy, Unsolved Problems in Number Theory, Section A1.

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

Cf. A007491, A013632, A053001, A014085, A070316, A085099, A058055, A069003.

[NextPrime(n^2) - n^2: n in [0..100]]; // Vincenzo Librandi, Jul 06 2015

Maple
```
A053000 := n->nextprime(n^2)-n^2;
```

nxt[n_]:=Module[{n2=n^2},NextPrime[n2]-n2]
nxt/@Range[0,100]  (* Harvey P. Dale, Dec 20 2010 *)

A053000(n)=nextprime(n^2)-n^2  \\ M. F. Hasler, Mar 23 2013

from sympy import nextprime
def a(n): nn = n*n; return nextprime(nn) - nn
print([a(n) for n in range(94)]) # Michael S. Branicky, Feb 17 2022

a(n) = A013632(n^2). - Robert Israel, Jul 06 2015

More terms from James Sellers, Feb 22 2000

A159829 a(n) is the smallest natural number m such that n^3+m^3+1^3 is prime.

Original entry on oeis.org

1, 2, 1, 2, 1, 4, 15, 2, 3, 2, 11, 10, 9, 2, 7, 14, 5, 4, 9, 2, 15, 2, 7, 16, 15, 8, 13, 2, 1, 10, 3, 4, 15, 2, 11, 10, 9, 2, 7, 6, 13, 22, 5, 2, 1, 6, 29, 10, 29, 10, 3, 2, 11, 12, 3, 8, 3, 2, 19, 6, 15, 8, 1, 2, 1, 18, 5, 2, 1, 18, 1, 12, 17, 14, 15, 26, 7, 6, 3, 2, 19, 12, 1, 18, 3, 8, 15, 2, 11, 6

Offset: 1

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Apr 23 2009

nonn

a(2k-1) is odd, a(2k) is even.
Exponent 2: There are infinitely many primes of the forms n^2+m^2 and n^2+m^2+1^2.
Exponent k>2: Are there infinitely many primes of the forms n^k+m^k and n^k+m^k+1^k?

			2^3+2^3+1=17 = A000040(7); a(2)=2.
7^3+15^3+1=3719 = A000040(519); a(7)=15.
21^3+15^3+1=18523 = A000040(2122), a(21)=15.

L. E. Dickson, History of the Theory of Numbers, Vol, I: Divisibility and Primality, AMS Chelsea Publ., 1999.
A. Weil, Number theory: an approach through history, Birkhäuser 1984.
David Wells, Prime Numbers: The Most Mysterious Figures in Math. John Wiley and Sons. 2005.

Michel Marcus, Table of n, a(n) for n = 1..10000

Cf. A069003, A159828.

Cf. A067200 (when m=1).

A159829 := proc(n) for m from 1 do if isprime(n^3+m^3+1) then RETURN(m) ; fi; od: end: seq(A159829(n),n=1..120) ; # R. J. Mathar, Apr 28 2009

snn[n_]:=Module[{n3=n^3,m=1},While[!PrimeQ[n3+1+m^3],m++];m]; Array[ snn,100] (* Harvey P. Dale, Sep 04 2019 *)

a(n) = my(m=1); while (!isprime(n^3+m^3+1^3), m++); m; \\ Michel Marcus, Nov 07 2023

Corrected and extended by R. J. Mathar, Apr 28 2009

A129634 Least nonnegative m such that T(n) + T(m) is prime, where T(n) = n*(n+1)/2.

Original entry on oeis.org

2, 1, 0, 1, 1, 7, 4, 1, 1, 7, 3, 1, 1, 3, 16, 13, 1, 4, 4, 1, 1, 4, 4, 1, 46, 3, 7, 1, 2, 7, 16, 2, 13, 4, 3, 1, 13, 3, 4, 22, 1, 16, 16, 1, 1, 7, 3, 1, 10, 3, 7, 1, 2, 7, 16, 2, 1, 4, 4, 13, 1, 4, 16, 1, 1, 16, 4, 2, 1, 16, 8, 1, 10, 3, 7, 1, 1, 31, 7, 2, 13, 4, 4, 10, 1, 8, 7, 13, 1, 43, 16, 5, 25, 16

Offset: 0

Jonathan Vos Post, May 31 2007

What is the simplest proof that this is defined for all nonzero n?

It appears that a(n)A130504 provides evidence that a(n) exists for all n. - T. D. Noe, Jun 04 2007

			a(6) = 4 because T(4) = 10 is the least triangular number whose sum with T(6) = 21 is prime, since {21+0 = 3*7, 21+3 = 2^3*3, 21+6 = 3^3} are all composite, but 21+10 = 31 is prime.

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

Cf. A000040, A000217, A129755, A130334.

Cf. A069003 (for square numbers).

nn=100; tri=Range[0,nn]Range[nn+1]/2; Table[k=1; While[k<=Length[tri] && !PrimeQ[tri[[k]]+tri[[n]]], k++ ]; If[k<=Length[tri], k-1,0], {n,Length[tri]}] (* T. D. Noe, Jun 04 2007 *)

a(n) = Min{m: m*(m+1)/2 + n*(n+1)/2 is prime}. a(n) = Min{m: A000217(m) + A000217(n) is an element of A000040}.

Corrected and extended by T. D. Noe, Jun 04 2007

A068486 Smallest prime equal to n^2 + m^2 with n >= m.

Original entry on oeis.org

2, 5, 13, 17, 29, 37, 53, 73, 97, 101, 137, 193, 173, 197, 229, 257, 293, 349, 397, 401, 457, 509, 593, 577, 641, 677, 733, 809, 857, 1021, 977, 1033, 1093, 1181, 1229, 1297, 1373, 1453, 1621, 1601, 1697, 1789, 1913, 2017, 2029, 2141, 2213, 2473, 2417, 2549

Offset: 1

Lekraj Beedassy, Mar 11 2002

nonn

With i being the imaginary unit, the numbers m + ni and m - ni are Gaussian primes. - Alonso del Arte, Feb 07 2011
All terms after the first are congruent to 1 (mod 4). - Carmine Suriano, Mar 30 2011
Any value can occur at most once (a consequence of Alonso del Arte's comment plus unique factorization in the Gaussian integers). - Robert Israel, Aug 19 2014
Smallest prime of the form (x^2 + y^2)/2 such that |x| + |y| = 2n. Note: |x| = n - m and |y| = n + m. - Thomas Ordowski and Altug Alkan, Jan 13 2017

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A068487. The values of m are given by A069003.

for n from 1 to 100 do m := 1:while(not isprime(n^2+m^2)) do m := m+1; end do:a[n] := n^2+m^2:end do:q := seq(a[i],i=1..100);

Table[k = 1; While[p = n^2 + k^2; Not[PrimeQ[p]], k++]; p, {n, 50}] (* Alonso del Arte, Feb 07 2011 *)

a(n) = for (m=1, n, if (isprime(p=n^2+m^2), return (p))); \\ Michel Marcus, Jan 22 2017

a(n) = n^2 + A069003(n)^2. - Thomas Ordowski, Aug 19 2014

More terms from Sascha Kurz, Mar 17 2002

A159828 a(n) is smallest number m > 0 such that m^2 + n^2 + 1 is prime.

Original entry on oeis.org

1, 6, 1, 6, 9, 2, 3, 6, 1, 6, 3, 2, 3, 6, 1, 6, 27, 8, 9, 24, 1, 6, 21, 4, 69, 12, 3, 6, 21, 6, 3, 6, 1, 6, 9, 2, 9, 6, 1, 6, 15, 6, 9, 6, 1, 6, 27, 2, 3, 36, 9, 6, 3, 6, 15, 18, 1, 48, 3, 4, 9, 6, 7, 6, 15, 4, 21, 42, 5, 6, 3, 2, 69, 18, 5, 6, 3, 2, 9, 24, 1, 6, 3, 8, 9, 6, 11, 18, 15, 4, 3, 6, 7, 18

Offset: 1

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Apr 23 2009

a(2k-1) is odd, a(2k) is even.

There are infinitely many primes of the forms n^2 + m^2 and n^2 + m^2 + 1, but it is not known if the number of primes of the form n^2 + 1 is infinite; cf. comments in A002496, A002313, A079544.

			n = 1: 1^2 + 1^2 + 1 = 3 is prime, so a(1) = 1.
n = 2: 1^2 + 2^2 + 1 = 6, 2^2 + 2^2 + 1 = 9, 3^2 + 2^2 + 1 = 14, 4^2 + 2^2 + 1 = 21, 5^2 + 2^2 + 1 = 30 are composite, but 6^2 + 2^2 + 1 = 41 is prime, so a(2) = 6.
n = 27: 1^2 + 27^2 + 1 = 731 = 17*43, 2^2 + 27^2 + 1 = 734 = 2*367 are composite, but 3^2 + 27^2 + 1 = 739 is prime, so a(27) = 3.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A069003 (smallest d such that n^2+d^2 is prime), A002496 (primes of form n^2+1), A002313 (primes of form x^2+y^2), A079544 (primes of form x^2+y^2+1, x>0, y>0).

S:=[]; for n in [1..100] do q:=n^2+1; m:=1; while not IsPrime(m^2+q) do m+:=1; end while; Append(~S,m); end for; S; // Klaus Brockhaus, May 21 2009

snm[n_]:=Module[{c=n^2+1,x=NextPrime[n^2+1]},While[!IntegerQ[Sqrt[x-c]], x= NextPrime[x]];Sqrt[x-c]]; Array[snm,100] (* Harvey P. Dale, Sep 22 2018 *)

Edited and extended by Klaus Brockhaus, May 21 2009

A227899 Number of primes p < n with 3*p - 4 and n^2 + (n - p)^2 both prime.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 1, 2, 1, 2, 2, 1, 3, 2, 1, 2, 2, 1, 2, 3, 2, 3, 3, 2, 3, 2, 3, 1, 1, 3, 2, 4, 2, 3, 3, 3, 4, 1, 2, 6, 2, 4, 2, 3, 5, 4, 2, 3, 4, 4, 4, 4, 2, 1, 2, 4, 2, 4, 2, 6, 7, 5, 3, 3, 9, 2, 3, 3, 2, 4, 4, 3, 1, 2, 8, 3, 6, 2, 2, 8, 4, 7, 2, 2, 5, 2, 3, 3, 2, 8, 3, 3, 1, 4, 7, 5, 9, 2, 2, 5

Offset: 1

Zhi-Wei Sun, Oct 14 2013

nonn

Conjecture: a(n) > 0 for all n > 3.

			a(5) = 1 since 5 = 3 + 2, and the three numbers 3, 3*3 - 4 = 5 and 5^2 + (5-3)^2 = 29 are all prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, preprint, arXiv:1211.1588.

Cf. A069003, A185636, A204065, A227898, A230223.

a[n_]:=Sum[If[PrimeQ[3Prime[i]-4]&&PrimeQ[n^2+(n-Prime[i])^2],1,0],{i,1,PrimePi[n-1]}]
Table[a[n],{n,1,100}]

A284211 a(n) is the least positive integer such that n^2 + a(n)^2 and n^2 + (a(n) + 2)^2 are primes.

Original entry on oeis.org

2, 1, 8, 9, 2, 29, 8, 3, 14, 1, 4, 23, 8, 9, 2, 29, 8, 5, 14, 1, 44, 13, 18, 59, 4, 9, 20, 13, 4, 11, 4, 3, 188, 9, 16, 149, 28, 13, 44, 1, 44, 23, 8, 19, 14, 19, 8, 35, 4, 17, 14, 3, 10, 59, 4, 9, 50, 3, 24, 29, 24, 43, 38, 9, 2, 89, 18, 5, 194, 17, 14, 5

Offset: 1

Lars-Erik Svahn, Mar 23 2017

nonn

z = n + i*a(n) and z' = n + i*(a(n) + 2) are two Gaussian primes such that |z - z'| = 2, corresponding to twin primes.

			a(1)=2: 1^2 + 1^2 = 2 is a prime but 1 + (1 + 2)^2 = 10 is not, while 1^2 + 2^2 = 5 and 1^2 + (2+2)^2 = 17 are both primes.

Robert Israel, Table of n, a(n) for n = 1..10000 (first 99 terms from Lars-Erik Svahn)
Lars-Erik Svahn, numbertheory.4th
Akshaa Vatwani, Bounded gaps between Gaussian primes, J. of Number Theory 171 (2017), 449-473.

Cf. A069003.

f:= proc(n) local k,pp,p;
    pp:= false;
    for k from (n+1) mod 2 by 2 do
      p:= isprime(n^2 + k^2);
      if p and pp then return k-2 fi;
      pp:= p;
    od;
end proc:
map(f, [$1..100]); # Robert Israel, Mar 30 2017

a[n_] := Block[{k = Mod[n, 2] + 1}, While[! PrimeQ[n^2 + k^2] || ! PrimeQ[n^2 + (k + 2)^2], k += 2]; k]; Array[a, 72] (* Giovanni Resta, Mar 23 2017 *)
lpi[n_]:=Module[{k=1},While[!AllTrue[n^2+{k^2,(k+2)^2},PrimeQ],k++];k]; Array[lpi,80] (* Harvey P. Dale, Aug 15 2024 *)

a(n) = my(k=n%2+1); while (!(isprime(n^2+k^2) && isprime(n^2+(k+2)^2)), k+=2); k  \\ Michel Marcus, Mar 25 2017

A284346 a(n) is the least positive integer such that n^2 + a(n)^2 and (n + 1)^2 + (a(n) + 1)^2 are primes.

Original entry on oeis.org

2, 1, 8, 1, 4, 1, 2, 3, 16, 3, 6, 7, 8, 1, 4, 1, 22, 5, 6, 3, 4, 17, 18, 5, 4, 1, 32, 5, 10, 29, 4, 27, 8, 15, 18, 1, 2, 15, 10, 3, 4, 247, 8, 15, 14, 19, 22, 35, 6, 19, 4, 27, 10, 11, 8, 1, 2, 5, 40, 13, 44, 127, 58, 61, 28, 1, 22, 13, 10, 19, 6, 7, 8, 15, 4, 9

Offset: 1

Lars-Erik Svahn, Mar 25 2017

nonn

n is odd iff a(n) is even.

			a(1)=2 since (1 + 1)^2 + (1 + 1)^2 is not prime, but 1^2 + 2^2 = 5 and (1 + 1)^2 + (2 + 1)^2 = 13 are prime.

Lars-Erik Svahn, Table of n, a(n) for n = 1..10000
Lars-Erik Svahn, numbertheory.4th
Akshaa Vatwani, Bounded gaps between Gaussian primes, Journal of Number Theory, Volume 171, February 2017, Pages 449-473.
Eric Weisstein's World of Mathematics, Gaussian Prime.
Index entries for Gaussian integers and primes.

Cf. A069003, A284211, A284327.

Rest@ FoldList[Module[{k = 1}, While[Times @@ Boole@ Map[PrimeQ, {#2^2 + k^2, (#2 + 1)^2 + (k + 1)^2}] < 1, k++]; k] &, 1, Range@ 76] (* Michael De Vlieger, Mar 25 2017 *)

a(n) = my(k=0); while (! (isprime(n^2+k^2) && isprime((n+1)^2+(k+1)^2)), k++); k;  \\ Michel Marcus, Mar 25 2017

A239388 Values of n such that n^2 + d^2 is prime for a record first value of d.

Original entry on oeis.org

1, 3, 8, 9, 12, 23, 30, 48, 63, 114, 141, 408, 651, 1173, 2697, 12639, 30963, 53343, 159537, 283209, 289131, 335511, 601398, 1832421, 2594214, 3533079, 4013361, 15717618, 17449677, 57532827, 186891843, 226385511, 231177657, 242117967

Offset: 1

T. D. Noe, Mar 18 2014

See A239389 for the corresponding values of d.

Cf. A069003 (least number d such that n^2 + d^2 is prime).

leastK[n_] := Module[{k = 1}, While[! PrimeQ[n^2 + k^2], k++]; k]; nn = 10000; t = {}; kMax = 0; Do[k = leastK[n]; If[k > kMax, kMax = k; AppendTo[t, {n, k}]], {n, nn}]; Transpose[t][[1]]

a(24)-a(31) from Giovanni Resta, Mar 18 2014

a(32)-a(34) from Emmanuel Vantieghem, Jan 13 2017

A239389 The record values of d for the n in A239388.

Original entry on oeis.org

1, 2, 3, 4, 7, 8, 11, 13, 32, 35, 40, 47, 50, 100, 118, 130, 178, 208, 220, 250, 254, 320, 353, 380, 401, 404, 466, 487, 598, 640, 652, 676, 680, 692

Offset: 1

T. D. Noe, Mar 18 2014

Cf. A069003 (least number d such that n^2 + d^2 is prime).

leastK[n_] := Module[{k = 1}, While[! PrimeQ[n^2 + k^2], k++]; k]; nn = 10000; t = {}; kMax = 0; Do[k = leastK[n]; If[k > kMax, kMax = k; AppendTo[t, {n, k}]], {n, nn}]; Transpose[t][[2]]

a(24)-a(31) from Giovanni Resta, Mar 18 2014

a(32)-a(34) from Emmanuel Vantieghem, Jan 13 2017

cp's OEIS Frontend

A053000 a(n) = (smallest prime > n^2) - n^2.

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A159829 a(n) is the smallest natural number m such that n^3+m^3+1^3 is prime.

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A129634 Least nonnegative m such that T(n) + T(m) is prime, where T(n) = n*(n+1)/2.

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A068486 Smallest prime equal to n^2 + m^2 with n >= m.

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A159828 a(n) is smallest number m > 0 such that m^2 + n^2 + 1 is prime.

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A227899 Number of primes p < n with 3*p - 4 and n^2 + (n - p)^2 both prime.

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A284211 a(n) is the least positive integer such that n^2 + a(n)^2 and n^2 + (a(n) + 2)^2 are primes.

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