cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 12 results. Next

A160022 Primes p such that p^4 + 5^4 + 3^4 is prime.

Original entry on oeis.org

3, 23, 47, 53, 67, 73, 89, 101, 103, 109, 151, 157, 179, 229, 521, 557, 569, 619, 661, 821, 977, 1013, 1087, 1129, 1277, 1321, 1451, 1559, 1607, 1627, 1741, 1867, 1871, 1949, 2137, 2389, 2441, 2797, 3271, 3313, 3643, 3677, 3769, 3847, 4001, 4027, 4133
Offset: 1

Views

Author

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Apr 30 2009

Keywords

Comments

For primes p, q, r the sum p^4 + q^4 + r^4 can be prime only if at least one of p, q, r equals 3. This sequence is the special case q = 5, r = 3.
It is conjectured that the sequence is infinite.
There are twin prime (101, 103) and other consecutive primes (151, 157; 1867, 1871) in the sequence.

Examples

			p = 3: 3^4 + 5^4 + 3^4 = 787 is prime, so 3 is in the sequence.
p = 5: 5^4 + 5^4 + 3^4 = 1331 = 11^3, so 5 is not in the sequence.
p = 101: 101^4 + 5^4 + 3^4 = 104061107 is prime, so 101 is in the sequence.
p = 103: 103^4 + 5^4 + 3^4 = 112551587 is prime, so 103 is in the sequence.

Crossrefs

Cf. A158979, A159829, A160031.

Programs

  • Magma
    [p: p in PrimesUpTo(5000)|IsPrime(p^4+706)] // Vincenzo Librandi, Dec 18 2010
  • Mathematica
    With[{c=5^4+3^4},Select[Prime[Range[600]],PrimeQ[#^4+c]&]] (* Harvey P. Dale, Aug 14 2011 *)
  • PARI
    is(n)=isprime(n) && isprime(n^4+706) \\ Charles R Greathouse IV, Jun 07 2016

Extensions

Edited, 1607 inserted and extended beyond 3643 by Klaus Brockhaus, May 03 2009

A160031 Primes p such that p^4 + 2*3^4 is prime.

Original entry on oeis.org

5, 13, 19, 43, 71, 83, 97, 101, 107, 109, 127, 149, 179, 193, 197, 211, 233, 241, 311, 353, 383, 401, 421, 541, 577, 599, 607, 619, 641, 647, 683, 709, 727, 751, 769, 827, 877, 883, 941, 967, 991, 1009, 1061, 1097, 1109, 1187, 1289, 1373, 1381, 1409, 1439
Offset: 1

Views

Author

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Apr 30 2009

Keywords

Comments

For primes p, q, r the sum p^4 + q^4 + r^4 can be prime only if at least one of p, q, r equals 3. This sequence is the special case q = r = 3.
It is conjectured that the sequence is infinite.
There are prime twins (107, 109) and other consecutive primes (193, 197) in the sequence.

Examples

			p = 5: 5^4 + 2*3^4 = 787 is prime, so 5 is in the sequence.
p = 7: 7^4 + 2*3^4 = 2563 = 11*233, so 7 is not in the sequence.
p = 107: 107^4 + 2*3^4 = 131079763 is prime, so 107 is in the sequence.
p = 109: 109^4 + 2*3^4 = 141158323 is prime, so 109 is in the sequence.

Links

Crossrefs

Cf. A158979, A159829, A160022.

Programs

Extensions

Edited and extended beyond 683 by Klaus Brockhaus, May 03 2009

A159082 Numbers whose squares added to 7! are prime.

Original entry on oeis.org

13, 23, 29, 59, 61, 73, 97, 101, 103, 109, 121, 127, 149, 169, 187, 191, 199, 221, 227, 251, 257, 263, 277, 299, 307, 317, 319, 331, 341, 367, 373, 383, 389, 397, 403, 407, 409, 433, 449, 451, 461, 463, 467, 491, 493, 499, 517, 527, 529, 533, 551, 563, 571
Offset: 1

Views

Author

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Apr 05 2009

Keywords

Comments

1) Necessarily a(n) is not divisible by 2, 3, 5, 7.
2) Sequence is conjectured to be infinite.
3) It is conjectured that an infinite number of terms are primes.
4) Note that sequence contains a(k), a(k+1) prime twin pairs, first are (59,61), (461,463), (827,829), (1319,1321).
5) It is conjectured that an infinite number of a(n) are squares, first are 121=11^2, 169=13^2, 529=23^2, 841=29^2, 961=31^2, 1681=41^2, ...
6) m!+k^2=n^2 are the generalized Brown number triples (m,k,n).

Examples

			1) 7!+1=71^2, (7, 71) is the largest (of three) Brown pairs; Erdos conjectured that there are no others.
2) 7!+3^2=5049= 3^3 * 11 * 17, 7!+5^2=5065 = 5 * 1013, 7!+7^2=5089 = 7 * 727, 7!+9^2=5121 = 3^2 * 569, 7!+11^2=5161 = 13 * 397.
3) 7!+13^2=5209 prime, so a(1)=13.

References

  • R. K. Guy, Unsolved Problems in Number Theory (2nd ed.) New York: Springer-Verlag, p. 193, 1994
  • I. Niven, H. S. Zuckerman and H. L. Montgomery: An Introduction to the Theory of Numbers (5th ed.). Wiley Text Books, 1991
  • David Wells, Prime Numbers: The Most Mysterious Figures in Math. John Wiley and Sons. 2005

Links

Crossrefs

Cf. A038202, A158979.

Programs

  • Mathematica
    With[{s = 7!}, Select[Range[600], PrimeQ[#^2 + s] &]] (* Harvey P. Dale, Jun 17 2015 *)
  • PARI
    isok(n) = isprime(n^2+7!); \\ Michel Marcus, Jul 23 2013; corrected Jun 14 2022

Formula

7! + a(n)^2 = prime.

Extensions

Edited by N. J. A. Sloane, Apr 05 2009

A160023 Primes p such that p^4 + 7^4 + 3^4 is prime.

Original entry on oeis.org

11, 37, 71, 101, 149, 163, 191, 271, 293, 379, 409, 419, 647, 661, 709, 1153, 1193, 1231, 1277, 1523, 1583, 1619, 1667, 1693, 1753, 1777, 1787, 1913, 2089, 2099, 2161, 2213, 2441, 2473, 2531, 2551, 2609, 2711, 2749, 2909, 2953, 2999, 3221, 3257, 3469
Offset: 1

Views

Author

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Apr 30 2009

Keywords

Comments

For primes p, q, r the sum p^4 + q^4 + r^4 can be prime only if at least one of p, q, r equals 3. This sequence is the special case q = 7, r = 3.
It is conjectured that the sequence is infinite.
There are prime twins (6197, 6199) and other consecutive primes (409, 419; 2089, 2099) in the sequence.

Examples

			p = 7: 7^4 + 7^4 + 3^4 = 4883 = 19*257, so 7 is not in the sequence.
p = 11: 11^4 + 7^4 + 3^4 = 17123 is prime, so 11 is in the sequence.
p = 101: 101^4 + 7^4 + 3^4 = 104062883 is prime, so 101 is in the sequence.

Links

Crossrefs

Cf. A158979, A159829, A160022.

Programs

  • Magma
    [ p: p in PrimesUpTo(3500) | IsPrime(p^4+2482) ]; // Klaus Brockhaus, May 03 2009
  • Mathematica
    Select[Prime[Range[500]],PrimeQ[#^4+2482]&] (* Harvey P. Dale, Jan 31 2017 *)

Extensions

Edited and extended beyond 2441 by Klaus Brockhaus, May 03 2009

A160024 Primes p such that p^4 + 11^4 + 3^4 is prime.

Original entry on oeis.org

7, 11, 13, 19, 23, 31, 41, 47, 61, 67, 73, 83, 101, 107, 127, 157, 163, 191, 193, 277, 281, 311, 337, 373, 379, 401, 409, 431, 443, 461, 491, 523, 541, 569, 607, 643, 673, 691, 719, 733, 743, 757, 769, 887, 929, 947, 953, 1031, 1039, 1087, 1093, 1097, 1103, 1109
Offset: 1

Views

Author

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Apr 30 2009

Keywords

Comments

For primes p, q, r the sum p^4 + q^4 + r^4 can be prime only if at least one of p, q, r equals 3. This sequence is the special case q = 11, r = 3.
It is conjectured that the sequence is infinite.
There are prime twins (11, 13) and other consecutive primes (7, 11; 1093, 1097) in the sequence.

Examples

			p = 3: 3^4 + 11^4 + 3^4 = 14803 = 113*131, so 3 is not in the sequence.
p = 7: 7^4 + 11^4 + 3^4 = 17123 is prime, so 7 is in the sequence.
p = 11: 11^4 + 11^4 + 3^4 = 29363 is prime, so 11 is in the sequence.
p = 13: 13^4 + 11^4 + 3^4 = 43283 is prime, so 13 is in the sequence.

Links

Crossrefs

Cf. A158979, A159829, A160022.

Programs

  • Magma
    [ p: p in PrimesUpTo(1110) | IsPrime(p^4+14722) ]; // Klaus Brockhaus, May 03 2009
  • Mathematica
    Select[Prime[Range[200]],PrimeQ[#^4+14722]&] (* Harvey P. Dale, Apr 18 2023 *)

Extensions

Edited and extended beyond 461 by Klaus Brockhaus, May 03 2009

A160025 Primes p such that p^4 + 13^4 + 3^4 is prime.

Original entry on oeis.org

3, 11, 13, 17, 31, 41, 43, 53, 83, 127, 167, 181, 193, 211, 241, 311, 337, 349, 421, 431, 487, 521, 557, 613, 617, 647, 701, 769, 811, 857, 953, 1021, 1151, 1249, 1289, 1303, 1373, 1453, 1459, 1471, 1523, 1553, 1567, 1579, 1613, 1663, 1669, 1747, 1823, 1831
Offset: 1

Views

Author

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Apr 30 2009

Keywords

Comments

For primes p, q, r the sum p^4 + q^4 + r^4 can be prime only if at least one of p, q, r equals 3. This sequence is the special case q = 13, r = 3.
It is conjectured that the sequence is infinite.
There are prime twins (11, 13) and other consecutive primes (421, 431; 1823, 1831) in the sequence.

Examples

			p = 3: 3^4 + 13^4 + 3^4 = 28723 is prime, so 3 is in the sequence.
p = 5: 5^4 + 13^4 + 3^4 = 29267 = 7*37*113, so 5 is not in the sequence.
p = 17: 17^4 + 13^4 + 3^4 = 112163 is prime, so 17 is in the sequence.
p = 83: 83^4 + 13^4 + 3^4 = 47486963 is prime, so 83 is in the sequence.

Links

Crossrefs

Cf. A158979, A159829, A160022.

Programs

  • Magma
    [ p: p in PrimesUpTo(1840) | IsPrime(p^4+28642) ]; // Klaus Brockhaus, May 03 2009
  • Mathematica
    Select[Prime[Range[400]],PrimeQ[#^4+28642]&] (* Harvey P. Dale, Dec 14 2011 *)

Extensions

Edited and extended beyond 857 by Klaus Brockhaus, May 03 2009

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

Views

Author

Derek Orr, Jul 08 2014

Keywords

Comments

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

Examples

			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.

Links

Crossrefs

Cf. A158979, A089489, A242553.

Programs

  • PARI
    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++)
  • Python
    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

Views

Author

Derek Orr, Jul 08 2014

Keywords

Comments

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

Examples

			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.

Links

Crossrefs

Cf. A158979, A089489, A242554.

Programs

  • PARI
    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++)
  • Python
    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

Views

Author

Derek Orr, Jul 08 2014

Keywords

Comments

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

Examples

			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.

Links

Crossrefs

Cf. A158979, A089489, A242555.

Programs

  • PARI
    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++)
  • Python
    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

Views

Author

Derek Orr, Jul 08 2014

Keywords

Comments

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

Examples

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

Links

Crossrefs

Cf. A158979, A089489, A242556.

Programs

  • PARI
    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++)
  • Python
    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
Showing 1-10 of 12 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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