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 1-7 of 7 results.

A072180 Numbers k such that 2^k - k^2 is prime.

Original entry on oeis.org

5, 7, 9, 17, 19, 51, 53, 81, 83, 119, 189, 219, 227, 301, 455, 461, 623, 2037, 2221, 2455, 3547, 5515, 6825, 8303, 9029, 12103, 49989, 55525, 64773, 80307, 119087, 141915, 192023, 205933, 301683, 307407
Offset: 1

Views

Author

Daniel Gronau (Daniel.Gronau(AT)gmx.de), Jun 30 2002

Keywords

Comments

The numbers corresponding to k = 2037, 2221, 3547 and 5515 have been certified prime with Primo. - Rick L. Shepherd, Nov 10 2002
The remaining k's > 1000 correspond only to probable primes.
Certainly k must be odd. Let N(k) = 2^k - k^2. Additional restrictions come from the facts that 7 | N(k) if k is in {2, 4, 5, 6, 10, 15} mod 21 and 17 | N(k) if k is in {31, 57, 61, 71, 107, 109, 113, 131} mod 136. - Daniel Gronau, Jul 06 2002
Henri Lifchitz found the terms > 40000 in 2001 and 119087 in March 2002. - Hugo Pfoertner, Nov 16 2004

Links

Crossrefs

Cf. A024012, A064539, A075896, A072164.

Programs

  • Mathematica
    Do[ If[ PrimeQ[ 2^n - n^2], Print[n]], {n, 1, 22850, 2}]
  • PARI
    is(n)=isprime(2^n-n^2) \\ Charles R Greathouse IV, Feb 17 2017

Extensions

Edited and extended by Robert G. Wilson v, Jul 01 2002
More terms from Hugo Pfoertner, Nov 16 2004
More terms from Henri Lifchitz submitted by Ray Chandler, Mar 02 2007

A072179 Numbers n such that n^(n+1) - (n+1)^n is prime.

Original entry on oeis.org

3, 6, 9, 12, 44, 883, 1277
Offset: 1

Views

Author

Daniel Gronau (Daniel.Gronau(AT)gmx.de), Jun 30 2002

Keywords

Comments

Some of the larger entries may only correspond to probable primes.
6279 is a term. - Alexander Adamchuk, Apr 09 2007

Crossrefs

Cf. A072164, A073499.

Programs

  • Mathematica
    Do[ If[ PrimeQ[ n^(n + 1) - (n + 1)^n], Print[n]], {n, 1, 2000}]
  • PARI
    is(n)=ispseudoprime(n^(n+1)-(n+1)^n) \\ Charles R Greathouse IV, Feb 17 2017

Extensions

Edited and extended by Robert G. Wilson v, Jul 02 2002

A243100 Primes of the form x^(y+1)-y^x, for x,y > 0.

Original entry on oeis.org

3, 7, 19, 179, 543607, 129136067, 94143168179, 11920928949924493, 36472996377170722403, 61159026180004467059, 1341068619659378429383, 10301051460877537453973547005699, 710542735760100185871124061615149, 17763568394002504646778106434649157
Offset: 1

Views

Author

M. F. Hasler, Aug 19 2014

Keywords

Comments

See A123206 for primes of the form x^y-y^x with x,y>1. If y=1 is allowed, any prime p is obtained for x=p+1; this motivates the "y+1" in the exponent of the present sequence.
See also A086877 (and A098463) for primes of the form (x+1)^x-x^x.
y=0 would give "Primes of the form x", so y>0 is required. y=1 gives x^2-1 = (x-1)*(x+1) which is only prime for x=2. - Jens Kruse Andersen, Aug 23 2014

Links

Crossrefs

See also A072164.

Programs

  • PARI
    a=[];for(S=1,199,for(x=1,S-1,ispseudoprime(p=x^(1+y=S-x)-y^x)&&a=concat(a,p)));vecsort(a) \\ The list calculated this way is probably not complete up to the last terms. E.g., a 46 digit prime is found for x=3, y=97 after three larger terms for smaller S=x+y.
  • PARI
    m=300; a=[]; for(x=1, m+5, for(y=1, m+5, p=x^(y+1)-y^x; if(p<2^m && ispseudoprime(p), a=concat(a, p)))); a=vecsort(a) \\ Compute all terms below 2^m. Jens Kruse Andersen, Aug 23 2014

A085682 Numbers k such that k^k - (k-1)^k is prime.

Original entry on oeis.org

2, 3, 7, 43, 79, 463, 1277
Offset: 1

Views

Author

Farideh Firoozbakht, Jul 17 2003

Keywords

Comments

Each term of the sequence must be prime.
a(8) > 30000. - Michael S. Branicky, Dec 06 2024

Examples

			7 is in the sequence because 7^7 - 6^7 = 543607 is prime.

Links

Crossrefs

Cf. A072164, A072179, A073499.

Programs

  • PARI
    isok(k) = ispseudoprime(k^k - (k-1)^k); \\ Jinyuan Wang, Mar 19 2020

A140669 Numbers k such that (k+1)^(k+1) - k^k is prime.

Original entry on oeis.org

1, 2, 3, 6, 10, 16, 105, 119, 1906, 7917
Offset: 1

Views

Author

Rick L. Shepherd, May 21 2008

Keywords

Comments

Terms found with PrimeForm. Primes corresponding to 16, 105 and 119 certified with Primo. 7917 corresponds to a 30870-digit probable prime.
a(11) > 20000. - Michael S. Branicky, Apr 11 2025

Examples

			17^17 - 16^16 = 808793517812627212561, which is prime, so 16 is a term.

Links

Crossrefs

Cf. A134985. Equals A072164 - 1.

Programs

A162591 Primes in A007781.

Original entry on oeis.org

3, 23, 229, 776887, 275311670611, 808793517812627212561, 47962816398523117606189726043968411848519304708598059350620557763277694737755820158580941773369740112983781265183299561695077810144494290292906506606685128216915382107158604900927276535058149770652889252352435564631
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Jul 07 2009

Keywords

Comments

See A072164 for a condensed representation of the same information.

Examples

			3^3-2^2=27-4=23 is prime and enters the list.

Crossrefs

Cf. A068955.

Programs

  • Mathematica
    f[n_]:=n^n-(n-1)^(n-1); lst={};Do[If[PrimeQ[f[n]],AppendTo[lst,f[n]]], {n,2,5!}];lst

Extensions

Definition simplified, reference to A072164 and A068955 added by R. J. Mathar, Aug 11 2009

A219978 Numbers k (>= 1) such that A007781(k-1) = k^k - (k-1)^(k-1) is semiprime.

Original entry on oeis.org

5, 6, 13, 16, 18, 21, 22, 28, 29, 37, 46, 60, 71, 84
Offset: 1

Views

Author

Jonathan Vos Post, Dec 02 2012

Keywords

Comments

This is to A072164 as semiprimes A001358 are to primes A000040. Can thus be called power difference semiprimes.
a(15) >= 115, as 115^115 - 114^114 is a 237-digit composite number with no known factors. - Tyler Busby, Feb 12 2023

Examples

			a(1) = 5 because 5^5 - 4^4 = 2869 = 19 * 151 is semiprime.
a(2) = 6 because 6^6 - 5^5 = 43531 = 101 * 431.
a(3) = 13 because 13^13 - 12^12 = 293959006143997 = 28201 * 10423708597.
a(4) = 16 because 16^16 - 15^15 = 18008850183328692241 = 109 * 165218809021364149.

Links

Crossrefs

Cf. A001358, A072164, A007781.

Programs

Formula

{ k : A007781(k-1) in A001358 }.

Extensions

a(9)-a(13) from Charles R Greathouse IV, Dec 02 2012
a(14) from Charles R Greathouse IV, Dec 04 2012
Showing 1-7 of 7 results.

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