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 1-8 of 8 results.

A273547 Primes of the form 2^(2^n) + 27.

Original entry on oeis.org

29, 31, 43, 283, 65563
Offset: 1

Views

Author

Vincenzo Librandi, May 31 2016

Keywords

Comments

Terms given correspond to n = 0, 1, 2, 3, and 4.
Next term >= 18.
Next term >= 2^2^22 + 27. - Charles R Greathouse IV, Jun 06 2016

Crossrefs

Cf. primes of the form 2^(2^n)+k: A160027 (k=15), this sequence (k=27), A160029 (k=51), A160028 (k=81), A160032 (k=93), A273548 (k=163), A273549 (k=165), A273550 (k=177), A273551 (k=253), A273552 (k=267), A273804 (k=301), A273805 (k=331), A273806 (k=357), A160030 (k=385), A273807 (k=427), A273808 (k=463), A273809 (k=487), A273810 (k=597), A160033 (k=757), A273811 (k=805), A160034 (k=807).

Programs

  • Magma
    [a: n in [0..15] | IsPrime(a) where a is 2^(2^n)+27];
  • Mathematica
    Select[Table[2^(2^n) + 27, {n, 0, 20}], PrimeQ]
  • PARI
    for(n=0,4, if(ispseudoprime(t=2^2^n+27), print1(t", "))) \\ Charles R Greathouse IV, Jun 06 2016

A160030 Primes of the form 2^(2^k)+385.

Original entry on oeis.org

389, 401, 641, 65921, 4294967681, 340282366920938463463374607431768211841
Offset: 1

Views

Author

Cino Hilliard, Apr 30 2009

Keywords

Comments

Terms given correspond to k= 1, 2, 3, 4, 5 and 7.
Next term >= 2^2^20 + 385. - Vincenzo Librandi, Jun 07 2016
Next term >= 2^2^33 + 385. - Charles R Greathouse IV, Jun 07 2016

Crossrefs

Cf. A019434, A160027, A160028, A160029, A160032, A160033, A160034.
Cf. similar sequences listed in A273547.

Programs

  • Magma
    [a: n in [0..15] | IsPrime(a) where a is 2^(2^n)+385]; // Vincenzo Librandi, Jun 07 2016
  • Mathematica
    Select[Table[2^(2^n) + 385, {n, 0, 20}], PrimeQ] (* Vincenzo Librandi, Jun 07 2016 *)
  • PARI
    g(n,m) = for(x=0,n,y=2^(2^x)+m;if(ispseudoprime(y),print1(y",")))

A160029 Primes of the form 2^(2^k)+51.

Original entry on oeis.org

53, 67, 307, 65587, 18446744073709551667, 340282366920938463463374607431768211507
Offset: 1

Views

Author

Cino Hilliard, Apr 30 2009

Keywords

Comments

Generated by k = 0, 2, 3, 4, 6, 7,...
a(7) > 2^(2^23)+51. - Martin Møller Skarbiniks Pedersen, May 31 2016

Crossrefs

Cf. A019434, A160027 - A160030.

Programs

  • Magma
    [a: n in [0..15] | IsPrime(a) where a is 2^(2^n)+51]; // Vincenzo Librandi, May 31 2016
  • Mathematica
    Select[Table[2^(2^n) + 51, {n, 0, 10}], PrimeQ] (* Vincenzo Librandi, May 31 2016 *)
  • PARI
    g(n,m) = for(x=0,n,y=2^(2^x)+m;if(ispseudoprime(y),print1(y",")))

A160032 Primes of the form 2^(2^k)+93.

Original entry on oeis.org

97, 109, 349, 65629, 4294967389, 18446744073709551709
Offset: 1

Views

Author

Cino Hilliard, Apr 30 2009

Keywords

Comments

Terms given correspond to k = 1, 2, 3, 4, 5 and 6.
Next term >= 2^2^24 + 93. - Charles R Greathouse IV, Jun 07 2016

Crossrefs

Cf. A019434, A160027, A160028, A160029, A160030, A160033, A160034.
Cf. similar sequences listed in A273547.

Programs

  • Magma
    [a: n in [0..15] | IsPrime(a) where a is 2^(2^n)+93]; // Vincenzo Librandi, Jun 07 2016
  • Maple
    select(isprime,[seq(2^(2^n)+93, n=0..15)]); # Robert Israel, Jun 07 2016
  • Mathematica
    Select[Table[2^(2^n) + 93, {n, 0, 15}], PrimeQ] (* Vincenzo Librandi, Jun 07 2016 *)
  • PARI
    g(n,m) = for(x=0,n,y=2^(2^x)+m;if(ispseudoprime(y),print1(y",")))

A160033 Primes of the form 2^(2^k)+757.

Original entry on oeis.org

761, 773, 1013, 66293, 18446744073709552373, 340282366920938463463374607431768212213, 115792089237316195423570985008687907853269984665640564039457584007913129640693
Offset: 1

Views

Author

Cino Hilliard, Apr 30 2009

Keywords

Comments

Terms given correspond to k= 1, 2, 3, 4, 6, 7 and 8.
Next term >= 2^2^26 + 757. - Charles R Greathouse IV, Jun 07 2016

Crossrefs

Cf. A019434, A160027-A160030, A160032-A160034.
Cf. similar sequences listed in A273547.

Programs

  • Magma
    [a: n in [0..10] | IsPrime(a) where a is 2^(2^n)+ 757]; // Vincenzo Librandi, Jun 05 2016
  • Mathematica
    Select[Table[2^(2^n) + 757, {n, 0, 10}], PrimeQ] (* Vincenzo Librandi, Jun 05 2016 *)
  • PARI
    g(n,m) = for(x=0,n,y=2^(2^x)+m;if(ispseudoprime(y),print1(y",")))

A160034 Primes of the form 2^(2^k) + 807 .

Original entry on oeis.org

809, 811, 823, 1063, 66343, 18446744073709552423, 115792089237316195423570985008687907853269984665640564039457584007913129640743
Offset: 1

Views

Author

Cino Hilliard, Apr 30 2009

Keywords

Comments

Terms given correspond to k= 0, 1, 2, 3, 4, 6 and 8.
Next term is >= 2^2^25 + 807. - Charles R Greathouse IV, Jun 07 2016

Crossrefs

Cf. A019434, A160027, A160028, A160029, A160030, A160032, A160033.
Cf. similar sequences listed in A273547.

Programs

  • Magma
    [ a: n in [0..10] | IsPrime(a) where a is 2^(2^n) + 807 ]; // Vincenzo Librandi, Jun 05 2016
  • Mathematica
    Select[Table[2^(2^n) + 807, {n, 0, 10}], PrimeQ] (* Vincenzo Librandi, Jun 05 2016 *)
  • PARI
    g(n,m) = for(x=0,n,y=2^(2^x)+m;if(ispseudoprime(y),print1(y",")))

A286680 Smallest nonnegative m such that (1 + n)^(2^m) + n is not prime.

Original entry on oeis.org

0, 5, 4, 2, 0, 3, 1, 0, 3, 3, 0, 1, 0, 0, 2, 4, 0, 0, 2, 0, 2, 1, 0, 2, 0, 0, 1, 0, 0, 2, 3, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 2, 0, 0, 0, 1, 0, 1, 1, 0, 3, 2, 0, 1, 0, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 1, 2, 0, 0, 0, 0, 1, 2, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 2
Offset: 0

Views

Author

Juri-Stepan Gerasimov, May 12 2017

Keywords

Comments

Nonprimes: 1, 4294967297, 43046723, 259, 9, 1679621, 55, 15, 43046729, 100000009, 21, 155, 25, 27, 50639, 18446744073709551631, 33, 35, ...
Conjecture: a(n) <= 6 for all n.
This conjecture would contradict the generalized Bunyakovsky conjecture. That is, the polynomials (1+n)^k+n for k=0..6 satisfy the conditions for that conjecture, and so there should be some n for which all seven are prime. - Robert Israel, May 17 2017
Smallest k such that (1 + k)^(2^n) + k is not prime: 0, 6, 3, 5, 2, 1, 54131988 (conjecturally finite). Last term found by Robert G. Wilson v, May 14 2017
From Robert G. Wilson v, May 18 2017: (Start)
m=
0: 0, 4, 7, 10, 12, 13, 16, 17, 19, 22, 24, 25, 27, 28, 31, 32, 34, 37, 38, etc.;
1: 6, 11, 21, 26, 33, 35, 36, 39, 41, 48, 50, 51, 56, 68, 74, 78, 81, 83, etc.;
2: 3, 14, 18, 20, 23, 29, 44, 54, 63, 65, 69, 75, 95, 99, 113, 114, 125, etc.;
3: 5, 8, 9, 30, 53, 119, 230, 308, 329, 350, 624, 638, 779, 785, 813, 1110, etc.;
4: 2, 15, 2100, 4223, 4773, 7868, 8744, 9339, 9540, 13178, 14589, 15884, etc.;
5: 1, 1432578, 1627035, 1737054, 1888094, 1959638, 2176139, 3172304, 3425069, etc.;
6: 54131988, 177386619, 229940778, 846372674, 2124404844, 2367307088, 2539775055, etc.;
(End)

Examples

			a(0) = 0 because (1 + 0)^(2^0) + 0 = 1 is not prime.

Links

Crossrefs

Cf. A019434, A047845, A057726, A160027.

Programs

  • Maple
    f:= proc(n) local k;
  for k from 0 while isprime((1+n)^(2^k)+n) do od:
  k;
end proc:
map(f, [$0..100]); # Robert Israel, May 17 2017
  • Mathematica
    f[n_] := Block[{k = 0}, While[ PrimeQ[(1 + n)^(2^k) + n], k++]; k]; Array[f, 105, 0] (* Robert G. Wilson v, May 14 2017 *)
  • PARI
    a(n) = {my(m = 0); while (isprime((1 + n)^(2^m) + n), m++); m;} \\ Michel Marcus, May 19 2017

A286982 Smallest nonnegative k such that (1 + k)^(2^n) + k is not prime and all (1 + k)^(2^j) + k, for 0 <= j < n, are primes.

Original entry on oeis.org

6, 3, 5, 2, 1, 54131988
Offset: 1

Views

Author

Juri-Stepan Gerasimov, May 12 2017

Keywords

Examples

			a(1) = 6 because (1 + 6)^(2^1) + 6 = 55 is semiprime and (1 + 6)^(2^0) + 6 = 13 is prime;
a(2) = 3 because (1 + 3)^(2^2) + 3 = 259 is semiprime and both (1 + 3)^(2^0) + 3 = 7 and (1 + 3)^(2^1) + 3 = 19 are primes;
a(3) = 5 because (1 + 5)^(2^3) + 5 = 167921 is semiprime and (1 + 5)^(2^0) + 5 = 11, (1 + 5)^(2^1) + 5 = 41 and (1 + 5)^(2^2) + 5 = 1301 are all primes;
a(4) = 2 because (1 + 2)^(2^4) + 2 = 43046723 is semiprime and (1 + 2)^(2^0) + 2 = 5, (1 + 2)^(2^1) + 2 = 11, (1 + 2)^(2^2) + 2 = 83 and (1 + 2)^(2^3) + 2 = 6563 are all primes;
a(5) = 1 because (1 + 1)^(2^5) + 1 = 4294967297 is semiprime and (1 + 1)^(2^0) + 1 = 3, (1 + 1)^(2^1) + 1 = 5, (1 + 1)^(2^2) + 1 = 17, (1 + 1)^(2^3) + 1 = 257 and (1 + 1)^(2^4) + 1 = 65537 are fix known Fermat primes (A019434);
a(6) = 54131988 because (1 + 54131988)^(2^6) + 54131988 is composite and (1 + 54131988)^(2^0) + 54131988 = 108263977, (1 + 54131988)^(2^1) + 54131988 = 2930272287228109, (1 + 54131988)^(2^2) + 54131988 =  8586495360054127683625679378629, (1 + 54131988)^(2^3) + 54131988 = 73727902568231063808600888120898279950965368674840612135914869, (1 + 54131988)^(2^4) + 54131988 and (1 + 54131988)^(2^5) + 54131988 are all primes.

Crossrefs

Cf. A019434, A047845, A057726, A160027, A286680.

Programs

  • Mathematica
    a[n_] := Block[{k = 1}, While[PrimeQ[(1 + k)^(2^n) + k] || ! AllTrue[(1 + k)^(2^Range[0, n-1]) + k, PrimeQ], k++]; k]; Array[a, 5] (* Giovanni Resta, May 30 2017 *)

Extensions

a(6) from Robert G. Wilson v, May 14 2017
Showing 1-8 of 8 results.

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