A069003 -id:A069003 - cp's OEIS frontend

A284327 a(n) is the least positive integer such that n^2 + a(n)^2 and n^2 + (a(n) - 2)^2 are primes.

1, 1, 10, 1, 4, 1, 10, 5, 16, 1, 6, 25, 10, 1, 4, 1, 10, 7, 16, 1, 46, 15, 20, 1, 6, 1, 22, 15, 6, 13, 6, 5, 190, 11, 18, 1, 30, 15, 46, 1, 46, 25, 10, 21, 16, 21, 10, 37, 6, 19, 16, 5, 12, 1, 6, 1, 52, 5, 26, 31, 26, 45, 40, 11, 4, 1, 20, 7, 196, 19, 16

Offset: 1

Lars-Erik Svahn, Mar 25 2017

nonn

n + i*a(n) and n + i*(a(n) - 2) are Gaussian twin primes.

If n^2 + 1 is a prime then a(n) = 1 else a(n) = A284211(n) + 2.

			a(1) = 1 since 1^2 + 1^2 = 2 and 1^2 + (1 - 2)^2 = 2 are primes.

Lars-Erik Svahn, Table of n, a(n) for n = 1..99
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. A005574, A069003, A284211, A284346.

Rest@ FoldList[Module[{k = 1}, While[Times @@ Boole@ Map[PrimeQ, {#2^2 + k^2, #2^2 + (k - 2)^2}] < 1, k++]; k] &, 1, Range@ 71] (* Michael De Vlieger, Mar 25 2017 *)

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

from sympy import isprime
def a(n):
    k=0
    while True:
        if isprime(n**2 + k**2) and isprime(n**2 + (k - 2)**2): return k
        else: k+=1
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Mar 31 2017

a(n) = 1 for n in A005574. - Michel Marcus, Mar 31 2017

A242553 Least number k such that n^8 + k^8 is prime.

Original entry on oeis.org

1, 1, 10, 1, 6, 5, 12, 13, 16, 3, 24, 7, 2, 3, 8, 9, 4, 17, 4, 7, 2, 3, 20, 7, 8, 19, 10, 3, 10, 19, 14, 17, 32, 11, 8, 25, 6, 25, 40, 7, 10, 43, 16, 5, 68, 7, 30, 5, 8, 19, 58, 17, 26, 17, 2, 11, 10, 3, 4, 49, 6, 71, 22, 15, 14, 47, 30, 9, 2, 19, 6, 19, 6, 5, 28, 13, 2

Offset: 1

Derek Orr, May 17 2014

nonn

If a(n) = 1, then n is in A006314.

			10^8+1^8 = 100000001 is not prime. 10^8+2^8 = 100000256 is not prime. 10^8+3^8 = 100006561 is prime. Thus, a(10) = 3.

Cf. A069003, A006314.

lnk[n_]:=Module[{c=n^8,k=1},While[CompositeQ[c+k^8],k++];k]; Array[lnk,80] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jan 12 2020 *)

a(n)=for(k=1,oo,if(ispseudoprime(n^8+k^8),return(k)));

import sympy
from sympy import isprime
def a(n):
  for k in range(10**4):
    if isprime(n**8+k**8):
      return k
n = 1
while n < 100:
  print(a(n))
  n += 1

A242554 Least number k such that n^16 + k^16 is prime.

Original entry on oeis.org

1, 1, 4, 3, 6, 5, 6, 7, 22, 13, 16, 5, 8, 5, 14, 11, 10, 7, 16, 31, 8, 9, 10, 11, 38, 29, 10, 9, 22, 61, 20, 5, 4, 3, 16, 11, 6, 25, 28, 7, 6, 17, 16, 1, 46, 9, 58, 61, 22, 41, 92, 3, 14, 19, 14, 23, 56, 37, 20, 109, 6, 121, 10, 39, 4, 67, 34, 11, 26, 9, 30, 11, 12, 1

Offset: 1

Derek Orr, May 17 2014

nonn

If a(n) = 1, then n is in A006313.

			4^16+1^16 = 4294967297 is not prime. 4^16+2^16 = 4295032832 is not prime. 4^16+3^16 = 4338014017 is prime. Thus, a(4) = 3.

Cf. A069003, A006313.

a(n)=for(k=1,oo,if(ispseudoprime(n^16+k^16),return(k)));

import sympy
from sympy import isprime
def a(n):
  for k in range(10**4):
    if isprime(n**16+k**16):
      return k
n = 1
while n < 100:
  print(a(n))
  n += 1

A242555 Least number k such that k^32 + n^32 is prime.

Original entry on oeis.org

1, 29, 40, 33, 34, 131, 50, 9, 8, 11, 10, 13, 12, 97, 166, 221, 200, 13, 10, 61, 176, 23, 22, 65, 94, 151, 352, 87, 2, 1, 38, 39, 4, 5, 48, 137, 18, 11, 4, 3, 60, 55, 40, 9, 106, 33, 10, 29, 134, 7, 44, 33, 50, 1, 38, 5, 148, 37, 2, 41, 10, 11, 94, 75, 4, 5, 100, 5, 22

Offset: 1

Derek Orr, May 17 2014

nonn

If a(n) = 1, then n is in A006315.

Cf. A069003, A006315.

lnk[n_]:=Module[{k=1,n32=n^32},While[!PrimeQ[n32+k^32],k++];k]; Array[ lnk,70] (* Harvey P. Dale, Apr 26 2018 *)

a(n)=for(k=1,oo,if(ispseudoprime(n^32+k^32),return(k)));

import sympy
from sympy import isprime
def a(n):
  for k in range(10**4):
    if isprime(n**32+k**32):
      return k
n = 1
while n < 100:
  print(a(n))
  n += 1

A242556 Least number k such that k^64 + n^64 is prime.

Original entry on oeis.org

1, 37, 32, 39, 118, 13, 16, 11, 154, 41, 8, 29, 6, 17, 64, 7, 14, 107, 66, 63, 58, 87, 38, 397, 282, 69, 32, 129, 12, 67, 210, 3, 200, 227, 82, 55, 2, 7, 4, 541, 10, 103, 64, 167, 286, 71, 60, 593, 6, 459, 14, 3, 2, 91, 4, 81, 98, 21, 164, 47, 36, 51, 10, 15, 84, 19, 30

Offset: 1

Derek Orr, May 17 2014

nonn

If a(n) = 1, then n is in A006316.

Cf. A069003, A006316.

lnk[n_]:=Module[{c=n^64,k=1},While[!PrimeQ[c+k^64],k++];k]; Array[lnk,70] (* Harvey P. Dale, Oct 21 2017 *)

a(n)=for(k=1,oo,if(ispseudoprime(n^64+k^64),return(k)));

import sympy
from sympy import isprime
def a(n):
  for k in range(10**4):
    if isprime(n**64+k**64):
      return k
n = 1
while n < 100:
  print(a(n))
  n += 1

A242557 Least number k such that n^128+k^128 is prime.

Original entry on oeis.org

1, 113, 106, 259, 304, 85, 212, 135, 158, 47, 62, 985, 84, 47, 518, 485, 178, 169, 106, 27, 88, 139, 632, 47, 44, 643, 20, 209, 606, 1529, 32, 31, 1094, 139, 754, 647, 38, 37, 262, 69, 94, 631, 90, 25, 38, 195, 10, 277, 232, 187, 554, 189, 10, 47, 216, 131, 1132, 173, 390

Offset: 1

Derek Orr, May 17 2014

nonn

If a(n) = 1, then n is in A056994.

Cf. A069003, A056994.

lnk[n_]:=Module[{c=n^128,k},k=If[EvenQ[c],1,2];While[!PrimeQ[c+ k^128],k = k+2];k]; Join[{1},Array[lnk,60,2]] (* Harvey P. Dale, Mar 17 2015 *)

a(n)=for(k=1,10^4,if(ispseudoprime(n^128+k^128),return(k)));
n=1;while(n<100,print(a(n));n+=1)

import sympy
from sympy import isprime
def a(n):
    for k in range(10**4):
        if isprime(n**128+k**128):
            return k
n = 1
while n < 100:
    print(a(n))
    n += 1

A284376 a(n) is the least nonnegative integer such that n + i*a(n) is a Gaussian prime.

Original entry on oeis.org

3, 1, 1, 0, 1, 2, 1, 0, 3, 4, 1, 0, 7, 2, 1, 2, 1, 2, 5, 0, 1, 4, 5, 0, 1, 4, 1, 2, 5, 4, 11, 0, 3, 2, 5, 2, 1, 2, 3, 10, 1, 4, 5, 0, 9, 2, 5, 0, 13, 4, 7, 4, 3, 10, 1, 4, 1, 2, 3, 0, 13, 10, 3, 32, 9, 2, 1, 0, 5, 10, 3, 0, 5, 2, 1, 4, 5, 10, 7, 0, 7, 4, 3, 0, 1, 2, 9, 2, 3, 4, 1, 4, 7, 8, 1, 2, 5, 2, 3, 4, 3

Offset: 0

Lars-Erik Svahn, Mar 25 2017

nonn

Lars-Erik Svahn, Table of n, a(n) for n = 0..10000
Lars-Erik Svahn, numbertheory.4th
Akshaa Vatwani, Bounded gaps between Gaussian primes, Journal of Number Theory, Volume 171, February 2017, Pages 449-473.
Wikipedia, Gaussian prime
Index entries for Gaussian integers and primes

Cf. A002145, A005574, A069003.

f:= proc(n) local k;
  for k from 0 do if GaussInt:-GIprime(n+I*k) then return k fi od
end proc:
map(f, [$0..100]); # Robert Israel, Apr 07 2017

Table[k = 0; While[! PrimeQ[n + I k, GaussianIntegers -> True], k++]; k, {n, 0, 100}] (* Michael De Vlieger, Mar 29 2017 *)

From Michel Marcus, Mar 30 2017: (Start)
a(n) = 0 for n in A002145.
a(n) = 1 for n in A005574.
(End)
a(n) = A069003(n) if n is not in A002145. - Robert Israel, Apr 07 2017

A242552 Least number k such that n^4 + k^4 is prime.

Original entry on oeis.org

1, 1, 2, 1, 2, 1, 2, 3, 2, 7, 2, 13, 4, 5, 8, 1, 2, 5, 2, 1, 10, 15, 2, 1, 6, 3, 2, 1, 12, 7, 12, 5, 14, 1, 6, 7, 2, 3, 14, 9, 2, 5, 10, 21, 2, 1, 4, 1, 2, 7, 2, 11, 6, 1, 14, 1, 2, 7, 2, 11, 2, 11, 8, 23, 16, 29, 12, 3, 10, 27, 2, 5, 8, 1, 8, 3, 20, 17, 2, 1, 10, 1, 10

Offset: 1

Derek Orr, May 17 2014

nonn

If a(n) = 1, then n is in A000068.

			8^4+1^4 = 4097 is not prime. 8^4+2^4 = 4112 is not prime. 8^4+3^4 = 4177 is prime. Thus, a(8) = 3.

Cf. A069003, A000068.

a(n)=for(k=1,oo,if(ispseudoprime(n^4+k^4),return(k)));

import sympy
from sympy import isprime
def a(n):
  for k in range(10**4):
    if isprime(n**4+k**4):
      return k
n = 1
while n < 100:
  print(a(n))
  n += 1

cp's OEIS Frontend

A284327 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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A242553 Least number k such that n^8 + k^8 is prime.

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A242554 Least number k such that n^16 + k^16 is prime.

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A242555 Least number k such that k^32 + n^32 is prime.

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A242556 Least number k such that k^64 + n^64 is prime.

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A242557 Least number k such that n^128+k^128 is prime.

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A284376 a(n) is the least nonnegative integer such that n + i*a(n) is a Gaussian prime.

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Maple

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A242552 Least number k such that n^4 + k^4 is prime.

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