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 1-5 of 5 results.

A057664 Primes of the form p + 2^p where p is prime.

Original entry on oeis.org

11, 37, 618970019642690137449562201, 266998379490113760299377713271194014325338065294581596243380200977777465722580068752870260867389
Offset: 1

Views

Author

Labos Elemer, Oct 16 2000

Keywords

Comments

The next term has 212 digits. - Harvey P. Dale, Feb 19 2018

Crossrefs

Programs

A057678 Primes of the form 2^p - p where p is prime.

Original entry on oeis.org

2, 5, 8179, 524269
Offset: 1

Views

Author

Labos Elemer, Oct 19 2000

Keywords

Comments

Next term, if it exists, has more than 618 digits. - Emeric Deutsch, Mar 27 2005
Next term, if it exists, has more than 10,000 digits.
The corresponding primes p are: 2, 3, 13, 19, .... - Gerasimov Sergey, Jul 26 2013
The corresponding 2^p - 1 are 3, 7, 8191, 524287 which are Mersenne primes (A000668). Is this the case for all members of the sequence? None of the other Mersenne primes < 2^132049-1 correspond to members of the sequence. - Robert Israel, Jul 18 2016
Next term is 2^481801-481801. 2^481801-1 is not a Mersenne prime. - Joerg Arndt, Jul 19 2016

Examples

			p=3 is prime, and so is 2^p - p = 8 - 3 = 5, so 5 is in the sequence. - _Michael B. Porter_, Jul 19 2016
		

Crossrefs

Programs

  • Maple
    a:=proc(n) if isprime(2^ithprime(n)-ithprime(n))=true then 2^ithprime(n)-ithprime(n) else fi end: seq(a(n),n=1..310); # Emeric Deutsch
  • Mathematica
    lst={};Do[p=Prime[n];If[PrimeQ[p=2^p-p],AppendTo[lst,p]],{n,5!}];lst (* Vladimir Joseph Stephan Orlovsky, Jan 28 2009 *)
    Select[Table[2^p-p,{p,Prime[Range[20]]}],PrimeQ] (* Harvey P. Dale, Sep 20 2018 *)

A057665 Numbers k such that prime(k) + 2^prime(k) is prime.

Original entry on oeis.org

2, 3, 24, 66, 126
Offset: 1

Views

Author

Labos Elemer, Oct 16 2000

Keywords

Comments

a(6) > 3002, since prime(3002) = 27479 (see comment by Ralf Stephan in A057663). - Michel Marcus, Dec 20 2013
These values correspond to indices ip = 2, 3, 9, 10, 11 where A052007(ip) is prime. - Michel Marcus, Dec 20 2013
If it exists, a(6) > 15*10^3. - Michal Paulovic, Sep 24 2023
If it exists, a(6) > 30*10^3. - Michael S. Branicky, Sep 01 2024

Crossrefs

Programs

  • Mathematica
    Select[Range[10^3], PrimeQ[# + 2^#] &@ Prime@ # &] (* Michael De Vlieger, Oct 26 2017 *)
  • PARI
    isok(n) = isprime(prime(n) + 2^prime(n)); \\ Michel Marcus, Dec 19 2013

Formula

a(n) = primepi(A057663(n)). - Michel Marcus, Dec 20 2013

Extensions

Name edited by Michel Marcus, Dec 19 2013

A295111 Primes p such that 2^p - p is also a prime.

Original entry on oeis.org

2, 3, 13, 19, 481801
Offset: 1

Views

Author

Iain Fox, Nov 14 2017

Keywords

Comments

a(6) > 1061095.
Intersection of A000040 and A048744.
Since numbers other than 3 that are congruent to 3 mod 6 are composite, for n > 2, a(n) is congruent to 1 mod 6 (see comments by Iain Fox in A048744).

Examples

			p=13, 2^13 - 13 = 8179 is prime.
		

Crossrefs

Programs

  • PARI
    lista(nn) = forprime(p=2, nn, if(ispseudoprime(2^p - p), print1(p, ", ")))

A383836 Integers k such that d*2^k + k/d is prime for some divisor d of k.

Original entry on oeis.org

1, 3, 5, 6, 9, 10, 15, 21, 22, 28, 39, 66, 75, 81, 89, 105, 108, 111, 141, 165, 166, 190, 196, 317, 340, 357, 459, 462, 483, 525, 564, 568, 573, 701, 735, 737, 792, 869, 1185, 1311, 1480, 1647, 1794, 1881, 2145, 2405, 2508, 2766, 3081, 3201, 3225, 3243, 4260, 4713, 5369, 5795, 5985
Offset: 1

Views

Author

Juri-Stepan Gerasimov, May 11 2025

Keywords

Examples

			6 is a term because 2*2^6 + 6/2 = 131 is prime for divisor d = 2 of k = 6.
		

Crossrefs

Supersequence of A057663.

Programs

  • Magma
    [k: k in [1..1000] | not #[d: d in Divisors(k) | IsPrime(d*2^k+(k div d))] eq 0];
    
  • Mathematica
    Select[Range[4300],Sum[Boole[PrimeQ[d*2^#+#/d]],{d,Divisors[#]}]>0 &] (* Stefano Spezia, May 16 2025 *)
  • PARI
    is(n, f=factor(n))=fordiv(n>>valuation(n,2),d, if(isprime(n/d*2^n+d), return(1))); 0 \\ Charles R Greathouse IV, May 17 2025

Extensions

a(41) corrected by Sean A. Irvine, May 21 2025
Showing 1-5 of 5 results.