A094133
Leyland primes: 3, together with primes of form x^y + y^x, for x > y > 1.
Original entry on oeis.org
3, 17, 593, 32993, 2097593, 8589935681, 59604644783353249, 523347633027360537213687137, 43143988327398957279342419750374600193, 4318114567396436564035293097707729426477458833, 5052785737795758503064406447721934417290878968063369478337
Offset: 1
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
A215434
Primes of form 2^k + k^2 - 1.
Original entry on oeis.org
2, 7, 31, 1123, 1180591620717411308323, 21778071482940061661655974875633165551139, 89202980794122492566142873090593446023942979, 1569275433846670190958947355801916604025588861116008664323
Offset: 1
-
[a: n in [0..250] | IsPrime(a) where a is 2^n+n^2-1];
-
Select[Table[2^n + n^2 - 1, {n, 0, 300}], PrimeQ]
A075896
Primes of the form 2^k - k^2.
Original entry on oeis.org
7, 79, 431, 130783, 523927, 2251799813682647, 9007199254738183, 2417851639229258349405791, 9671406556917033397642519, 664613997892457936451903530140158127, 784637716923335095479473677900958302012794430558004278391
Offset: 1
-
[a: n in [1..200] | IsPrime(a) where a is 2^n-n^2]; // Vincenzo Librandi, Dec 07 2011
-
Select[Table[2^n - n^2, {n, 500}], PrimeQ] (* Vincenzo Librandi, Dec 07 2011 *)
-
for(n=2,10^7,if(isprime(2^n-n^2),print1(2^n-n^2",")))
A064539
Numbers n such that 2^n + n^2 is prime.
Original entry on oeis.org
1, 3, 9, 15, 21, 33, 2007, 2127, 3759, 29355, 34653, 57285, 99069, 1933695
Offset: 1
- J.-M. De Koninck & A. Mercier, 1001 Problemes en Theorie Classique Des Nombres, Problem 165 pp. 30, 160, Ellipses Paris 2004.
a(14) from
Ryan Propper, May 11 2023. n=1933695 corresponds to a probable prime with 582101 digits, and was PRP tested with PFGW.
A173907
Primes of form x^y+y^x where x and y are composite numbers.
Original entry on oeis.org
43143988327398957279342419750374600193, 5052785737795758503064406447721934417290878968063369478337, 205688069665150755269371147819668813122841983204711281293004769, 3329896365316142756322307042065269797678257903507506764421250291562312417, 814539297859635326656252304265822609649892589675472598580095801187688932052096060144958129
Offset: 1
The first 5 terms are 15^32+32^15, 33^38+38^33, 8^69+69^8, 9^76+76^9, 21^68+68^21.
-
N:= 10^100: # for terms <= N
R:= NULL:
for x from 4 while 2*x^x < N do
if isprime(x) then next fi;
for y from x+1 do
if igcd(x,y) > 1 or isprime(y) then next fi;
q:= x^y + y^x;
if q > N then break fi;
if isprime(q) then R:= R,q fi;
od od:
sort([R]); # Robert Israel, Jul 11 2025
A182329
Primes of the form 5^n + n^2.
Original entry on oeis.org
29, 641, 15661, 244140769, 55511151231257827021181583404541018541
Offset: 1
-
Select[Table[5^n + n^2, {n, 0, 500}], PrimeQ] (* T. D. Noe, Apr 25 2012 *)
A173909
Numbers n such that prime(n) can be expressed as x+y in at least one way such that x^y + y^x is prime and 1 < x <= y.
Original entry on oeis.org
3, 5, 7, 9, 10, 15, 17, 18, 20, 24, 29, 32, 39, 42, 47, 55, 57, 62
Offset: 1
3 is in the sequence because 2^3 + 3^2 is prime and 2+3 = 5 = 3rd prime;
5 is in the sequence because 2^9 + 9^2 is prime and 2+9 = 11 = 5th prime;
7 is in the sequence because 2^15 + 15^2 is prime and 2+15 = 17 = 7th prime;
9 is in the sequence because 2^21 + 21^2 is prime and 2+21 = 23 = 9th prime;
10 is in the sequence because 5^24 + 24^5 is prime and 5+24 = 29 = 10th prime.
Constraint "0
Jon E. Schoenfield (after comments from R. J. Mathar regarding missing terms and from Wolfdieter Lang noting that the existing definition would make this sequence identical to A000027), Apr 12 2014
A182328
Primes of the form 4^n + n^2.
Original entry on oeis.org
5, 73, 1049, 16433, 67109033, 1237940039285380274899126249
Offset: 1
-
Select[Table[4^n + n^2, {n, 0, 200}], PrimeQ] (* T. D. Noe, Apr 25 2012 *)
A182357
Primes of the form 2^n + n^2 + 2.
Original entry on oeis.org
3, 5, 19, 59, 179, 8363, 131363, 134218459, 2147484611, 49039857307708443467467104868809893875799651909875303859, 3291009114642412084309938365114701009965471731267159726697262571
Offset: 1
-
Select[Table[2^n+n^2+2,{n,1000}],PrimeQ] (* Harvey P. Dale, Jul 22 2012 *)
Showing 1-9 of 9 results.
Comments