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.

Previous Showing 21-25 of 25 results.

A125957 Numbers n such that (2^n + 11^n)/13 is prime.

Original entry on oeis.org

3, 5, 17, 67, 83, 101, 1373, 6101, 12119, 61781
Offset: 1

Views

Author

Alexander Adamchuk, Feb 06 2007

Keywords

Comments

All terms are primes. Corresponding primes of the form (2^n + 11^n)/13 are {103, 12391, 38880540653801911, ...}.
a(11) > 10^5. - Robert Price, Feb 26 2013

Crossrefs

Cf. A000978 = numbers n such that (2^n + 1)/3 is prime. Cf. A057469 = numbers n such that (2^n + 3^n)/5 is prime. Cf. A082387 = numbers n such that (2^n + 5^n)/7 is prime.

Programs

  • Mathematica
    Do[p=Prime[n];f=(2^p+11^p)/13; If[PrimeQ[f], Print[{p, f}]], {n, 1, 100}]
  • PARI
    is(n)=ispseudoprime((2^n+11^n)/13) \\ Charles R Greathouse IV, Feb 20 2017

Extensions

2 more terms from Ryan Propper, Feb 09 2008
a(9)-a(10) from Robert Price, Feb 26 2013

A125958 Least number k > 0 such that (2^k + (2n-1)^k)/(2n+1) is prime.

Original entry on oeis.org

3, 3, 3, 5, 3, 3, 7, 3, 5, 5, 11, 3, 19, 11, 3, 229, 47, 5, 257, 3, 19, 31, 17, 11, 13, 3, 3, 5, 5, 59, 23, 3, 3, 7, 79, 3, 3373, 3, 3, 7, 13, 7, 7, 3527, 593, 19, 3, 3, 13, 13, 11, 19, 41, 3, 7, 109, 3, 227, 13, 5, 5, 3, 239, 5, 3251, 3, 1237, 3, 7, 31, 3, 7
Offset: 1

Views

Author

Alexander Adamchuk, Feb 06 2007

Keywords

Comments

All terms are odd primes.
a(38),...,a(43) = {3,3,7,13,7,7}.
a(46),...,a(64) = {19,3,3,13,13,11,19,41,3,7,109,11,227,13,5,5,3,239,5}.
a(66) = 3. a(68),...,a(72) = {3,7,31,3,7}.
a(74),...,a(92) = {3,5,19,17,3,83,3,3,19,19,11,11,61,3,7,7,3,41,29}.
a(94) = 5. a(97),a(98) = {19,7}. a(100) = 31.
a(n) is currently unknown for n = {37,44,45,65,67,73,93,95,96,99,...}.
From Kevin P. Thompson, May 18 2022: (Start)
All known terms from n=1..100 correspond to proven primes.
a(96) > 10250.
a(99) > 10250. (End)
Presuming every prime is seen at least once, one can specifically seek those with fixed k. Doing this, a(174) = 37, a(368) = 43 for example. - Bill McEachen, Aug 26 2024

Examples

			For n=4, the expression (2^k + (2n-1)^k)/(2n+1) takes on values 1, 53/9, 39, 2417/9, and 1871 for k=1..5. Since 1871 is the first prime number to occur, a(4) = 5.

Links

Crossrefs

Cf. A000978 ((2^n + 1)/3 is prime), A057469 ((2^n + 3^n)/5 is prime).
Cf. A082387 ((2^n + 5^n)/7 is prime), A125955 ((2^n + 7^n)/9 is prime).
Cf. A125956 ((2^n + 9^n)/11 is prime), A125955 ((2^n + 11^n)/13 is prime).

Programs

  • Mathematica
    Do[k = 1; While[ !PrimeQ[(2^k + (2n-1)^k)/(2n+1)], k++ ]; Print[k], {n, 100}] (* Ryan Propper, Mar 29 2007 *)

Extensions

More terms from Ryan Propper, Mar 29 2007
a(65)-a(72) from Kevin P. Thompson, May 18 2022

A227979 Integers not of the form (a^k+b^k)/(a+b) for any positive integer values of a, b, k with b > a.

Original entry on oeis.org

2, 4, 6, 8, 9, 14, 16, 18, 22, 23, 24, 32, 33, 36, 38, 42, 44, 46, 47, 54, 56, 59, 62, 64, 66, 69, 71, 72, 77, 81, 83, 86, 88, 92, 94, 96, 98, 99, 107, 114, 118, 121, 126, 128, 131, 132, 134, 138, 141, 142, 144, 152, 154, 158, 161, 162, 166, 167, 168, 177
Offset: 1

Views

Author

Robert Price, Sep 30 2013

Keywords

Comments

This form, (a^k+b^k)/(a+b), is a generalization of the Fermat numbers.
Not all integers are in this set.
See A229791 for the complement of this sequence.

Links

Crossrefs

A few of the sequences using this form that identify primes are A000978, A007658, A057469, A128066, A057171, A082387, A122853, A128335.

Programs

  • Mathematica
    limit=200; lst = {}; Do[p = (a^k + b^k)/(a + b); If[p <= limit && IntegerQ[p], AppendTo[lst, p]], {k, Log[2,3*limit+1]}, {b, 2, limit*2}, {a, b-1}]; Complement[Range[limit], Union[lst]]

A229791 Integers generated by (a^k+b^k)/(a+b) for all possible positive integer values of a,b,k with b>a.

Original entry on oeis.org

1, 3, 5, 7, 10, 11, 12, 13, 15, 17, 19, 20, 21, 25, 26, 27, 28, 29, 30, 31, 34, 35, 37, 39, 40, 41, 43, 45, 48, 49, 50, 51, 52, 53, 55, 57, 58, 60, 61, 63, 65, 67, 68, 70, 73, 74, 75, 76, 78, 79, 80, 82, 84, 85, 87, 89, 90, 91, 93, 95, 97, 100, 101, 102, 103
Offset: 1

Views

Author

Robert Price, Sep 29 2013

Keywords

Comments

This form, (a^k+b^k)/(a+b), is a generalization of the Fermat numbers.
Not all integers are in this set.
See A227979 for the complement of this sequence.

Links

Crossrefs

A few of the sequences using this form that identify primes are A000978, A007658, A057469, A128066, A057171, A082387, A122853, A128335.

Programs

  • Mathematica
    limit=105; lst = {}; Do[p = (a^k + b^k)/(a + b); If[p <= limit && IntegerQ[p], AppendTo[lst, p]], {k, Log[2,3*limit+1]}, {b, 2, limit*2}, {a, b-1}]; Union[lst]

A125955 Numbers k such that (2^k + 7^k)/9 is prime.

Original entry on oeis.org

5, 23, 73, 101, 401, 419, 457, 811, 1163, 1511, 8011
Offset: 1

Views

Author

Alexander Adamchuk, Feb 06 2007

Keywords

Comments

All terms are primes. Corresponding primes of the form (2^k + 7^k)/9 are {1871, 3040971926676589439, 5469081705798319217773539465593130845206220817280793349743311, ...}.
a(12) > 10^5. - Robert Price, Aug 28 2012

Crossrefs

Cf. A000978 = numbers n such that (2^n + 1)/3 is prime. Cf. A057469 = numbers n such that (2^n + 3^n)/5 is prime. Cf. A082387 = numbers n such that (2^n + 5^n)/7 is prime.

Programs

  • Mathematica
    Do[p=Prime[n];f=(2^p+7^p)/9; If[PrimeQ[f], Print[{p, f}]], {n, 1, 1000}]
  • PARI
    is(n)=ispseudoprime((2^n+7^n)/9) \\ Charles R Greathouse IV, Jun 13 2017

Extensions

More terms from Ryan Propper, Mar 23 2007
Previous Showing 21-25 of 25 results.

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