A049423 -id:A049423 - cp's OEIS frontend

A059844 a(n) = smallest nonzero square x^2 such that n+x^2 is prime.

1, 1, 4, 1, 36, 1, 4, 9, 4, 1, 36, 1, 4, 9, 4, 1, 36, 1, 4, 9, 16, 1, 36, 49, 4, 81, 4, 1, 144, 1, 16, 9, 4, 9, 36, 1, 4, 9, 4, 1, 576, 1, 4, 9, 16, 1, 36, 25, 4, 9, 16, 1, 36, 25, 4, 81, 4, 1, 324, 1, 36, 9, 4, 9, 36, 1, 4, 81, 4, 1, 36, 1, 16, 9, 4, 25, 36, 1, 4, 9, 16, 1, 144, 25, 4, 81

Offset: 1

Labos Elemer, Feb 26 2001

nonn

a(n) = 1 for n in A006093. - Robert Israel, Dec 31 2023

			a(24) = 49 because 49 + 24 = 73 is prime and 1 + 24 = 25, 4 + 24 = 28, 9 + 24 = 33, 16 + 24 = 40, 25 + 24 = 49, and 36 + 24 = 60 are composite.

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

Cf. A002496, A006093, A056899, A049423, A005473, A056905, A056909.

f:= proc(n) local x;
 for x from 1 + (n mod 2) by 2  do
  if isprime(n+x^2) then return x^2 fi;
 od
end proc:
f(1):= 1:
map(f, [$1..100]); # Robert Israel, Dec 31 2023

sqs[n_]:=Module[{q=1},While[!PrimeQ[n+q],q=(Sqrt[q]+1)^2];q]; Array[ sqs,90] (* Harvey P. Dale, Aug 11 2017 *)

a(n) + n is the smallest prime of the form x^2 + n.

A089124 Largest prime factor of n^2 + 3.

Original entry on oeis.org

2, 7, 3, 19, 7, 13, 13, 67, 7, 103, 31, 7, 43, 199, 19, 37, 73, 109, 13, 31, 37, 487, 19, 193, 157, 97, 61, 787, 211, 43, 241, 79, 13, 61, 307, 433, 7, 1447, 127, 229, 421, 31, 463, 277, 13, 163, 79, 769, 601, 2503, 31, 2707, 37, 139, 757, 73, 271, 37, 67, 1201, 19

Offset: 1

Cino Hilliard, Dec 05 2003

H. Rademacher, Lectures on Elementary Number Theory, pp. 33-38.

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

Cf. A049423, A006530, A117950.

FactorInteger[#][[-1,1]]&/@(Range[70]^2+3) (* Harvey P. Dale, Jun 17 2020 *)

largeasqp3(m) = { for(a=1,m, y=a^2 + 3; f = factor(y); v = component(f,1); v1 = v[length(v)]; print1(v1",")) }

a(n) = A006530(A117950(n)). - Jason Yuen, Aug 25 2024

A309844 Primes of the form n^4 + n^2 + 3.

Original entry on oeis.org

3, 5, 23, 653, 10103, 83813, 160403, 234743, 280373, 1049603, 3420653, 6252503, 11319863, 52207853, 92246423, 146422103, 174913853, 221548343, 442071653, 479807123, 577224653, 607597853, 655385603, 937921253, 1222865933, 1249233683, 1387525253, 1506177293

Offset: 1

Christopher R. Madan, Aug 19 2019

Digital root of all values > 3 is 5, compare A017221.

Subset of A027753. Subset of A017221.

Cf. A037896, A182344, A182345, A049423.

a = [];
for n = 0:1e3
    x = n.^4+n.^2+3;
    if isprime(x); a = [a,x]; end;
end

f[n_] := n^4 + n^2 + 3; Select[f /@ Range[0, 200], PrimeQ] (* Amiram Eldar, Aug 24 2019 *)

from sympy import isprime
a = []
for n in range(0,1000):
    x = n**4+n**2+3
    if isprime(x):
        a.append(x)

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A059844 a(n) = smallest nonzero square x^2 such that n+x^2 is prime.

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A089124 Largest prime factor of n^2 + 3.

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A309844 Primes of the form n^4 + n^2 + 3.

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