A064539 -id:A064539 - cp's OEIS frontend

A094133 Leyland primes: 3, together with primes of form x^y + y^x, for x > y > 1.

3, 17, 593, 32993, 2097593, 8589935681, 59604644783353249, 523347633027360537213687137, 43143988327398957279342419750374600193, 4318114567396436564035293097707729426477458833, 5052785737795758503064406447721934417290878968063369478337

Offset: 1

Lekraj Beedassy, May 04 2004

nonn

Contains A061119 as a subsequence.

			2^1 + 1^2, 3^2 + 2^3, 9^2 + 2^9, 15^2 + 2^15, 21^2 + 2^21, 33^2 + 2^33, 24^5 + 5^24, 56^3 + 3^56, 32^15 + 15^32, 54^7 + 7^54, 38^33 + 33^38.

Charles R Greathouse IV and Hans Havermann (Charles R Greathouse IV to 49), Table of n, a(n) for n = 1..100
Ed Copeland and Brady Haran, Leyland Numbers, Numberphile video (2014).
Hans Havermann, Table of n (where known), Leyland index, number of digits in decimal representation, and (x,y) pair for all known solutions.
Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.
Paul Leyland, Primes and PRPs of the form x^y + y^x.
Norman Luhn, Leyland table, 1st kind.

Cf. A061119 (primes where one of x,y is 2), A064539 (non-2 values where one of x,y is 2), A253471 (non-3 values where one of x,y is 3), A073499 (subset listing y where x = y+1), A076980 (Leyland numbers).

N:= 10^100: # to get all terms <= N
A:= {3}:
for n from 2 while 2*n^n < N do
  for k from n+1 do if igcd(n,k)=1 then
     a:= n^k + k^n;
     if a > N then break fi;
     if isprime(a) then A:= A union {a} fi fi;
  od
od:
A; # if using Maple 11 or earlier, uncomment the next line
# sort(convert(A,list)); # Robert Israel, Apr 13 2015

a = {3}; Do[Do[k = m^n + n^m; If[PrimeQ[k], AppendTo[a, k]], {m, 2, n}], {n, 2, 100}]; Union[a] (* Artur Jasinski *)
Prepend[Flatten[Map[Function[n, Map[Function[m, If[PrimeQ[m^n + n^m], m^n + n^m, Sequence[], Nothing]], Range[2, n]]], Range[2, 50]], 1], 3]//Union (* Mikk Heidemaa, Mar 27 2025 *)

f(x)=my(L=log(x)); L/lambertw(L) \\ finds y such that y^y == x
list(lim)=my(v=List()); for(x=2,f(lim/2), my(y=x+1,t); while((t=x^y+y^x)<=lim, if(ispseudoprime(t), listput(v,t)); y+=2)); Set(v) \\ Charles R Greathouse IV, Oct 28 2014

Corrected and extended by Jens Kruse Andersen, Oct 26 2007

Edited by Hans Havermann, Apr 10 2015

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

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

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

Henri Lifchitz, Renaud Lifchitz, PRP Top Records 2^n-n^2.

Cf. A024012, A064539, A075896, A072164.

Do[ If[ PrimeQ[ 2^n - n^2], Print[n]], {n, 1, 22850, 2}]

is(n)=isprime(2^n-n^2) \\ Charles R Greathouse IV, Feb 17 2017

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

A061119 Primes in the sequence n^2 + 2^n (A001580).

Original entry on oeis.org

3, 17, 593, 32993, 2097593, 8589935681

Offset: 1

Amarnath Murthy, Apr 21 2001

nonn

p and p^2 + 2^p are both prime only for p=3. All positive n satisfy the congruence n=3 (mod 6). - Lekraj Beedassy, Sep 07 2004
For values of n, see A064539. - Lekraj Beedassy, Jan 01 2007
The next term has 605 digits. - Harvey P. Dale, Jul 19 2017

			a(3) = 593 = 2^9 + 9^2.
a(4) = 32993 = 2^15 + 15^2.

J.-M. De Koninck & A. Mercier, 1001 Problemes en Theorie Classique Des Nombres, Problem 165 pp. 30; 160, Ellipses Paris 2004.

Subsequence of A094133.

Cf. A001580, A064539, A075896.

Select[Table[n^2+2^n,{n,1000}],PrimeQ] (* Harvey P. Dale, Jul 19 2017 *)

for(n=1,10^7, if(isprime(n^2+2^n),print(n^2+2^n)))

a(n) = A001580(A064539(n)). - Elmo R. Oliveira, Feb 18 2025

More terms from Jason Earls, Aug 09 2001. Next term too large to include.

A109216 Smallest factor of 2^(2n+1)+(2n+1)^2.

Original entry on oeis.org

3, 17, 3, 3, 593, 3, 3, 32993, 3, 3, 2097593, 3, 3, 73, 3, 3, 8589935681, 3, 3, 17, 3, 3, 11, 3, 3, 83, 3, 3, 11, 3, 3, 17, 3, 3, 857, 3, 3, 71329, 3, 3, 59, 3, 3, 19, 3, 3, 11, 3, 3, 17, 3, 3, 4561633, 3, 3, 2940725100673, 3, 3, 193, 3, 3, 1867, 3, 3, 17, 3, 3, 44449, 3, 3, 19, 3, 3

Offset: 0

Zak Seidov, Jun 24 2005

For n={0, 1, 4, 7, 10, 16, 1003, 1063, 1879, 14677, 17326, 28642, 49534} we have the only primes of the form 2^(2n+1)+(2n+1)^2: A064539. Cases 2n+1 == 1, 2 (mod 3) are trivial and may be omitted.

Vincenzo Librandi, Table of n, a(n) for n = 0..100

Cf. A064539, A109215.

Table[ With[ {k=2}, FactorInteger[ n^k + k^n]] [[1, 1]], {n, 1, 145, 2}]

More terms from Robert G. Wilson v, Jun 24 2005

A253471 Numbers k such that 3^k + k^3 is prime.

Original entry on oeis.org

2, 56, 10112, 63880, 78296, 125330, 222748, 1839730

Offset: 1

Michel Lagneau, Jan 01 2015

All terms == 2 or 4 mod 6. - Robert Israel, Jan 01 2015

			2 is in the sequence because 3^2 + 2^3 = 17 is prime.
56 is in the sequence because 3^56 + 56^3 = 523347633027360537213687137 is prime.

Cf. A001585 (3^n + n^3), A064539 (2^n + n^2 is prime), A094133 (Leyland primes).

select(t -> isprime(3^t+t^3), [seq(seq(6*i+j, j=[2,4]), i=0..100)]); # Robert Israel, Jan 01 2015

Do[If[PrimeQ[3^n+n^3], Print[n]], {n, 0, 12000}]

is(n)=ispseudoprime(3^n+n^3) \\ Charles R Greathouse IV, Jun 06 2017

a(4)-a(7) from Hans Havermann, Apr 30 2015

a(8) from Ryan Propper, Jun 27 2023

A276203 Numbers k such that k^9 + 9^k is prime.

Original entry on oeis.org

2, 76, 122, 422, 2300, 5090, 7166, 58046, 91382, 234178, 314738

Offset: 1

Felix Fröhlich, Aug 27 2016

Numbers k such that A185277(k) is prime.

			2 is a term of the sequence, because A185277(2) = 593 is prime.

Hans Havermann, Indices of A094133 in A076980.
Andrey V. Kulsha, Factorizations of x^y + y^x for 1 < y < x < 151.
P. Leyland, Primes and Strong Pseudoprimes of the form x^y + y^x.

Cf. A064539, A185277, A253471.

PARI
```
is(n) = ispseudoprime(n^9+9^n)
```

cp's OEIS Frontend

A094133 Leyland primes: 3, together with primes of form x^y + y^x, for x > y > 1.

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

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A061119 Primes in the sequence n^2 + 2^n (A001580).

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A109216 Smallest factor of 2^(2n+1)+(2n+1)^2.

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A253471 Numbers k such that 3^k + k^3 is prime.

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Maple

Mathematica

PARI

Extensions

A276203 Numbers k such that k^9 + 9^k is prime.

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PARI