A089489 -id:A089489 - cp's OEIS frontend

A158979 a(n) is the smallest number > n such that n^4 + a(n)^4 is prime.

2, 3, 4, 5, 8, 7, 10, 9, 10, 13, 16, 13, 14, 15, 22, 17, 20, 23, 24, 29, 38, 29, 26, 41, 26, 27, 28, 33, 34, 37, 32, 37, 34, 35, 52, 37, 38, 39, 46, 41, 50, 53, 44, 47, 58, 55, 50, 49, 60, 61, 62, 61, 56, 55, 58, 59, 68, 61, 62, 73, 66, 77, 64, 67, 84, 71

Offset: 1

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Apr 01 2009

For exponent 2 instead of 4 see A089489: Pythagorean triple has a prime hypotenuse.
Corresponding sequences with odd exponent u are impossible: x^u + y^u has factor x+y.
a(2k-1) is even, a(2k) is odd, a(n)-n is odd.
Conjecture: a(n) exists for all n, i.e., the sequence is well-defined and infinite.
Conjecture: a(n)-n = 1 for infinitely many n.
The largest value of a(n)-n for n <= 100 occurs at n = 90: 121-90 = 31.
a(n)-n = 1 for 35 values of n <= 100.

			1^4 + 2^4 = 17 is prime, so a(1) = 2.
2^4 + 3^4 = 97 is prime, so a(2) = 3.
5^4 + 6^4 = 1921 = 17*113, 5^4 + 7^4 = 3026 = 2*17*89, 5^4 + 8^4 = 4721 is prime, so a(5) = 8.

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

Cf. A089489.

S:=[]; for n in [1..72] do q:=n^4; k:=n+1; while not IsPrime(q+k^4) do k+:=1; end while; Append(~S, k); end for; S; // Klaus Brockhaus, Apr 12 2009

sn[n_]:=Module[{k=n+1,n4=n^4},While[CompositeQ[n4+k^4],k++];k]; Array[sn,80] (* Harvey P. Dale, Aug 09 2023 *)

Edited and entries verified by Klaus Brockhaus, Apr 12 2009

Corrected by Harvey P. Dale, Aug 09 2023

A068487 Smallest prime equal to n^2 + m^2 with n

Original entry on oeis.org

5, 13, 73, 41, 61, 157, 113, 233, 181, 269, 317, 313, 569, 421, 709, 617, 613, 853, 761, 929, 1117, 1013, 1429, 1201, 1301, 1637, 1753, 1873, 1741, 1861, 2897, 2113, 2689, 2381, 2521, 3697, 2969, 3469, 3121, 3449, 3617, 3613, 4153, 4337, 4729, 4517, 4513

Offset: 1

Lekraj Beedassy, Mar 11 2002

nonn

All terms are congruent to 1 (mod 4). - Carmine Suriano, Mar 30 2011

Carmine Suriano, Table of n, a(n) for n = 1..872

Cf. A089489 (values of m).

for n from 1 to 100 do m := n+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 = n + 1; While[p = n^2 + k^2; ! PrimeQ[p], k++]; p, {n, 100}] (* T. D. Noe, Mar 30 2011 *)

More terms from Sascha Kurz, Mar 17 2002

A378898 a(n) is the least k > 0 such that (n+k)^2 + n^2 is prime.

Original entry on oeis.org

1, 1, 5, 1, 1, 5, 1, 5, 1, 3, 3, 1, 7, 1, 7, 3, 1, 5, 1, 3, 5, 1, 7, 1, 1, 5, 5, 5, 1, 1, 13, 1, 7, 1, 1, 13, 3, 7, 1, 3, 3, 1, 5, 5, 7, 3, 1, 5, 25, 1, 5, 5, 5, 5, 3, 5, 11, 5, 5, 1, 3, 3, 17, 7, 1, 5, 13, 27, 1, 1, 13, 1, 27, 5, 19, 9, 3, 5, 1, 9, 19, 1, 5, 1, 1, 9, 1, 15, 7, 1, 3, 3, 5, 5, 7

Offset: 1

Robert Israel, Dec 11 2024

nonn

			a(3) = 5 because (3+5)^2 + 3^2 = 73 is prime, and no smaller number works.

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

Cf. A027861 (a(n) = 1), A089489, A378945, A378946.

f:= proc(n) local k;
  for k from n+1 by 2 do
    if igcd(k,n) = 1 and isprime(k^2 + n^2) then return k-n fi
  od
end proc;
map(f, [$1..100]);

a(n) = my(k=1); while (!isprime((n+k)^2 + n^2), k++); k; \\ Michel Marcus, Dec 11 2024

a(n) = A089489(n) - n.

A244932 Least number k > n such that k^8 + n^8 is prime.

Original entry on oeis.org

2, 13, 10, 17, 6, 37, 12, 13, 16, 27, 24, 71, 16, 31, 64, 43, 18, 43, 26, 23, 32, 29, 24, 79, 32, 53, 34, 61, 92, 47, 40, 33, 34, 57, 36, 47, 40, 53, 40, 79, 44, 43, 68, 91, 68, 57, 66, 61, 60, 53, 58, 83, 60, 91, 94, 61, 82, 61, 70, 101, 82, 71, 68, 145, 82, 67, 76, 69, 100

Offset: 1

Derek Orr, Jul 08 2014

nonn

a(n) = n+1 iff n is in A153504.

			13^8 + 14^8 = 2291519777 is not prime, 13^8 + 15^8 = 3378621346 is not prime. 13^8 + 16^8 = 5110698017 is prime. Thus a(13) = 16.

Jens Kruse Andersen, Table of n, a(n) for n = 1..10000

Cf. A158979, A089489, A242553.

a(n)=for(k=n+1,10^4,if(isprime(k^8+n^8),return(k)))
n=1;while(n<100,print1(a(n),", ");n++)

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

A244947 Least number k > n such that k^16 + n^16 is prime.

Original entry on oeis.org

2, 85, 4, 33, 6, 7, 8, 13, 22, 13, 16, 41, 28, 15, 22, 19, 24, 23, 54, 31, 22, 27, 56, 61, 38, 29, 40, 37, 34, 61, 32, 35, 82, 35, 46, 43, 40, 49, 58, 67, 42, 55, 58, 49, 46, 61, 58, 61, 68, 73, 92, 63, 94, 77, 166, 57, 82, 63, 72, 109, 76, 121, 82, 79, 86, 67, 72, 77, 82, 71, 98

Offset: 1

Derek Orr, Jul 08 2014

nonn

a(n) = n+1 iff n is in A154535.

			10^16 + 11^16 = 55949729863572161 is not prime. 10^16 + 12^16 = 2^16*(5^16+6^16) is not prime. 10^16 + 13^16 = 675416609183179841 is prime. Thus a(10) = 13.

Jens Kruse Andersen, Table of n, a(n) for n = 1..10000

Cf. A158979, A089489, A242554.

a(n)=for(k=n+1,10^4,if(isprime(k^16+n^16),return(k)))
n=1;while(n<100,print1(a(n),", ");n++)

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

A244948 Least number k > n such that k^32 + n^32 is prime.

Original entry on oeis.org

30, 29, 40, 33, 34, 131, 50, 9, 44, 11, 38, 13, 18, 97, 166, 221, 200, 37, 82, 61, 176, 23, 102, 65, 94, 151, 352, 87, 38, 37, 38, 39, 46, 37, 48, 137, 54, 55, 68, 43, 60, 55, 146, 51, 106, 87, 82, 65, 134, 53, 106, 103, 90, 71, 96, 71, 148, 91, 94, 139, 74, 69, 94, 75, 86, 169, 100

Offset: 1

Derek Orr, Jul 08 2014

nonn

a(n) = n+1 iff n is in A174156.

			34^32 + 35^32 = 35884563485651241417769982593434627193100364196481 is not prime. 34^2 + 36^32 = 2^32*(17^32+18^32) is not prime. 34^32 + 37^32 = 162384303092765940334383766635859112907663593431937 is prime. Thus a(34) = 37.

Jens Kruse Andersen, Table of n, a(n) for n = 1..10000

Cf. A158979, A089489, A242555.

a(n)=for(k=n+1,10^4,if(isprime(k^32+n^32),return(k)))
n=1;while(n<100,print1(a(n),", ");n++)

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

A244949 Least number k > n such that k^64 + n^64 is prime.

Original entry on oeis.org

102, 37, 32, 39, 118, 13, 16, 11, 154, 41, 94, 29, 158, 17, 64, 291, 70, 107, 66, 63, 58, 87, 38, 397, 282, 69, 32, 129, 142, 67, 210, 87, 200, 227, 82, 55, 70, 137, 388, 541, 140, 103, 64, 167, 286, 71, 60, 593, 262, 459, 62, 69, 92, 91, 128, 81, 98, 149, 164, 107, 192, 103

Offset: 1

Derek Orr, Jul 08 2014

nonn

a(n) = n+1 iff n is in A174157.

			8^64 + 11^64 = 4457915690803004131256192897205630962697827851093882159977969339137 is prime. Since 8^64 + 10^64 and 8^64 + 9^64 are both composite, a(8) = 11.

Jens Kruse Andersen, Table of n, a(n) for n = 1..10000

Cf. A158979, A089489, A242556.

a(n)=for(k=n+1,10^4,if(isprime(k^64+n^64),return(k)))
n=1;while(n<100,print1(a(n),", ");n++)

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

A244950 Least number k > n such that k^128 + n^128 is prime.

Original entry on oeis.org

120, 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, 194, 209, 606, 1529, 32, 113, 1094, 139, 754, 647, 38, 45, 262, 69, 94, 631, 90, 527, 326, 195, 54, 277, 232, 187, 554, 189, 78, 799, 216, 131, 1132, 173

Offset: 1

Derek Orr, Jul 08 2014

nonn

a(n) = n+1 iff n is in A215431.

			The n-value for which n^128 + 1 is prime (sequence A056994) is n = 120 (where n > 1 by definition). Thus a(1) = 120.

Jens Kruse Andersen, Table of n, a(n) for n = 1..1000

Cf. A158979, A089489, A242557.

lnk[n_]:=Module[{k=n+1,n128=n^128},While[!PrimeQ[n128+k^128],k++];k]; Array[lnk,60] (* Harvey P. Dale, Apr 22 2018 *)

a(n)=for(k=n+1,10^4,if(isprime(k^128+n^128),return(k)))
n=1;while(n<100,print1(a(n),", ");n++)

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

cp's OEIS Frontend

A158979 a(n) is the smallest number > n such that n^4 + a(n)^4 is prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

Extensions

A068487 Smallest prime equal to n^2 + m^2 with n

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Extensions

A378898 a(n) is the least k > 0 such that (n+k)^2 + n^2 is prime.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

PARI

Formula

A244932 Least number k > n such that k^8 + n^8 is prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Python

A244947 Least number k > n such that k^16 + n^16 is prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Python

A244948 Least number k > n such that k^32 + n^32 is prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Python

A244949 Least number k > n such that k^64 + n^64 is prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Python

A244950 Least number k > n such that k^128 + n^128 is prime.