A076980 -id:A076980 - 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

A051442 a(n) = n^(n+1)+(n+1)^n.

Original entry on oeis.org

1, 3, 17, 145, 1649, 23401, 397585, 7861953, 177264449, 4486784401, 125937424601, 3881436747409, 130291290501553, 4731091158953433, 184761021583202849, 7721329860319737601, 343809097055019694337, 16248996011806421522977

Offset: 0

N. J. A. Sloane

Odd prime p divides a(p-2). For n>1, a(prime(n)-2)/prime(n) = A125074(n) = {1, 29, 3343, 407889491, 298572057493, 454195874136455153, ...}. Prime p divides a((p+5)/2) for p = {19, 23, 61}. - Alexander Adamchuk, Nov 18 2006
From Mathew Englander, Jul 08 2020: (Start)
For all n != 1, a(n) mod 8 = 1.
If n mod 6 = 0, 3, or 5, then a(n) mod 6 = 1. If n mod 6 = 1, then a(n) mod 6 = 3. If n mod 6 = 2 or 4, then a(n) mod 6 = 5.
For all n, a(n)-1 is a multiple of n^2.
For n odd  and n >= 3, a(n)-1 is a multiple of (n+1)^2.
For n even and n >= 0, a(n)+1 is a multiple of (n+1)^2.
For proofs, see the Englander link. (End)

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Mathew Englander, Notes on OEIS A051442

Cf. A007925, A051443, A125074, A076980, A173054, A208506.

[n^(n+1)+(n+1)^n: n in [0..20]]; // Vincenzo Librandi, Jan 12 2012

Table[n^(n+1)+(n+1)^n,{n,0,20}] (* Harvey P. Dale, Oct 02 2018 *)

A051442[n]:=n^(n+1)+(n+1)^n$ makelist(A051442[n],n,0,30); /* Martin Ettl, Oct 29 2012 */

a(n)=(n+1)^n+n^(n+1) \\ Charles R Greathouse IV, Jan 12 2012

a(n) = (n + e + o(1)) * n^n. - Charles R Greathouse IV, Jan 12 2012
From Mathew Englander, Jul 08 2020: (Start)
a(n) = A093898(n+1, n) for n >= 1.
a(n) = a(n-1) + A258389(n) for n >= 1.
a(n) = A007778(n) + A000169(n+1).
(End)

A173054 Numbers of the form x^y + y^x, 1 < x < y.

Original entry on oeis.org

17, 32, 57, 100, 145, 177, 320, 368, 593, 945, 1124, 1649, 2169, 2530, 4240, 5392, 7073, 8361, 16580, 18785, 20412, 23401, 32993, 60049, 65792, 69632, 94932, 131361, 178478, 262468, 268705, 397585, 423393, 524649, 533169, 1048976, 1058576

Offset: 1

Vladimir Joseph Stephan Orlovsky, Feb 08 2010

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

T. D. Noe, Table of n, a(n) for n=1..1000

Cf. A076980, A385232, A385614.

f[a_,b_]:=a^b+b^a; Take[Union[Flatten[Table[f[a,b],{a,2,50},{b,a+1,50}]]],80]
nn=10^50; n=1; Union[Reap[While[n++; k=n+1; num=n^k+k^n; num

A055652 Table T(m,k)=m^k+k^m (with 0^0 taken to be 1) as square array read by antidiagonals.

Original entry on oeis.org

2, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 8, 4, 1, 1, 5, 17, 17, 5, 1, 1, 6, 32, 54, 32, 6, 1, 1, 7, 57, 145, 145, 57, 7, 1, 1, 8, 100, 368, 512, 368, 100, 8, 1, 1, 9, 177, 945, 1649, 1649, 945, 177, 9, 1, 1, 10, 320, 2530, 5392, 6250, 5392, 2530, 320, 10, 1, 1, 11, 593, 7073

Offset: 0

Henry Bottomley, Jun 08 2000

Columns and rows are A000012 (apart from first term), A000027, A001580, A001585, A001589, A001593 etc. Diagonals include A013499 (apart from first two terms), A051442, A051489.
Cf. A055651.
Contribution from Franklin T. Adams-Watters, Oct 26 2009: (Start)
Rows/columns: A000027, A001580, A001585, A001589, A001593, A001594, A001596.
Main diagonal is 2 * A000312. More diagonals: A051442, A051489, A155539.
Cf. A076980, A156353, A156354. (End)

E.g.f. Sum(n,m, T(n,m)/(n! m!)) = e^(x e^y) + e^(y e^x). [From Franklin T. Adams-Watters, Oct 26 2009]

A045575 Nonnegative numbers of the form x^y - y^x, for x,y > 1.

Original entry on oeis.org

0, 1, 7, 17, 28, 79, 118, 192, 399, 431, 513, 924, 1844, 1927, 2800, 3952, 6049, 7849, 8023, 13983, 16188, 18954, 32543, 58049, 61318, 61440, 65280, 130783, 162287, 175816, 255583, 261820, 357857, 523927, 529713, 1038576, 1048176

Offset: 1

Erich Friedman

Pillai proved that there are ~ 0.5 * (log x)^2/(log log x)^2 terms of this sequence up to x. - Charles R Greathouse IV, Jul 20 2017

Conjecture: For d > 11, 10^d - d^10 is the largest (base-ten) d-digit term. - Hans Havermann, Jun 12 2023

S. S. Pillai, On the indeterminate equation x^y - y^x = a, Journal Annamalai University 1, Nr. 1, (1932), pp. 59-61. Cited in Waldschmidt 2009.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Michel Waldschmidt, Perfect Powers: Pillai's works and their developments, arXiv:0908.4031 [math.NT], 2009.

Cf. A076980.

N:= 10^8: # to get all terms <= N
A:= (0,1):
for x from 2 while x^(x+1) - (x+1)^x <= N do
   for y from x+1 do
      z:= x^y - y^x;
      if z > N then break
      elif z > 0 then A:=A,z;
      fi
od od:
{A}; # Robert Israel, Aug 20 2014

Union[Flatten[Table[If[a^b-b^a>-1&&a^b-b^a<10^6*2,a^b-b^a],{a,1,123},{b,a,144}]]] (* Vladimir Joseph Stephan Orlovsky, Apr 26 2008 *)
nn=10^50; n=1; Union[Reap[While[n++; k=n+1; num=Abs[n^k-k^n]; num

list(lim)=my(v=List([0]),t); for(x=2,max(logint(lim\=1,2)+1,6), for(y=2,x-1, t=abs(x^y-y^x); if(t<=lim&&t, listput(v,t)))); Set(v) \\ Charles R Greathouse IV, Jul 20 2017

A155539 a(n) = n^(n+3) + (n+3)^n.

Original entry on oeis.org

1, 5, 57, 945, 18785, 423393, 10609137, 292475249, 8804293473, 287589316833, 10137858491849, 383799398752905, 15536767912476993, 669920208810550337, 30659724555890596833, 1484638520651877849057, 75846305139481944586817

Offset: 0

Vladimir Joseph Stephan Orlovsky, Jan 23 2009

nonn

1^4 + 4^1 = 5, 2^5 + 5^2 = 57, ...

Vincenzo Librandi, Table of n, a(n) for n = 0..300

Cf. A046065, A051442, A007925, A051489, A153217, A076980.

[n^(n+3)+(n+3)^n: n in [0..20] ]; // Vincenzo Librandi, Aug 25 2011

lst={};Do[m=n+3;q=n^m+m^n;AppendTo[lst,q],{n,0,4!}];lst
Table[n^(n+3)+(n+3)^n,{n,0,20}] (* Harvey P. Dale, Aug 18 2024 *)

Offset corrected by Arkadiusz Wesolowski, Aug 24 2011

A173055 Numbers of the form a^b+b^a, a and b are odd primes, b > a.

Original entry on oeis.org

368, 2530, 94932, 178478, 1596520, 48989176, 129145076, 1162268326, 1221074418, 1996813914, 94143190994, 96951758924, 762940872982, 19073488804224, 36314872537968, 68630377389272, 232630924325880, 617673396313738

Offset: 1

Vladimir Joseph Stephan Orlovsky, Feb 08 2010

nonn

3^5 + 5^3 = 368, 3^7 + 7^3 = 2530, 5^7 + 7^5 = 94932,..

Cf. A076980, A173054

f[a_,b_]:=Prime[a]^Prime[b]+Prime[b]^Prime[a]; Take[Union[Flatten[Table[f[a,b],{a,2,60},{b,a+1,60}]]],40]

A386892 Numbers k expressible as x^y + y^z + z^x, where x, y, and z are integers > 1.

Original entry on oeis.org

12, 21, 36, 44, 61, 81, 89, 104, 105, 166, 172, 181, 276, 288, 289, 324, 395, 401, 480, 597, 673, 768, 773, 777, 932, 972, 1065, 1128, 1230, 1250, 1376, 1905, 2033, 2089, 2173, 2244, 2545, 2557, 3182, 3388, 3493, 4148, 4244, 4368, 4393, 4652, 4774

Offset: 1

Ian Hahus, Aug 06 2025

nonn

			a(7) = 89, which can be given by x=4, y=3, z=2.

Andrew Howroyd, Table of n, a(n) for n = 1..10000

Cf. A123207 (subsequence of primes).

Cf. A076980.

upto(lim) = { my(L=List()); for(x=2, logint(lim,2), for(y=2, min(x,logint(lim,x)), for(z=2, min(x,logint(lim,y)), my(t=x^y+y^z+z^x); if(t<=lim, listput(L,t)) ))); Set(L) } \\ Andrew Howroyd, Aug 06 2025

A173056 Numbers of the form a^b+b^a, a and b are primes.

Original entry on oeis.org

8, 17, 54, 57, 177, 368, 2169, 2530, 6250, 8361, 94932, 131361, 178478, 524649, 1596520, 1647086, 8389137, 48989176, 129145076, 536871753, 1162268326, 1221074418, 1996813914, 2147484609, 94143190994, 96951758924, 137438954841

Offset: 1

Vladimir Joseph Stephan Orlovsky, Feb 08 2010

nonn

2^2+2^2=8, 2^3+3^2=17, 3^3+3^3=54, 2^5+5^2=57, 2^7+7^2=177, 3^5+5^3=368,..

Cf. A076980, A173054, A173055.

nMax=10^12; lim=PrimePi[Log[2, nMax]]; f[a_,b_]:=Prime[a]^Prime[b] + Prime[b]^Prime[a]; Select[Union[Flatten[Table[f[a,b], {a,lim}, {b,lim}]]], #<=nMax&]
nMax=10^12; lim=PrimePi[Log[2, nMax]]; Select[Union[First[#]^Last[#] + Last[#]^First[#]&/@ Tuples[Prime[Range[lim]],{2}]], #<=nMax&]  (* Harvey P. Dale, Mar 12 2011 *)

A160814 a(1) = 1; a(n+1) = a(n)^n + n^a(n).

Original entry on oeis.org

1, 2, 8, 7073

Offset: 1

Vladimir Joseph Stephan Orlovsky, May 26 2009

nonn

Next term is too large to display: 2.3459495195697547514*10^4258

The next term (a(5)) has 4259 digits. - Harvey P. Dale, Jul 18 2021

Cf. A093898, A055652, A076980. - R. J. Mathar, May 29 2009

a=1;lst={};Do[a=a^n+n^a;AppendTo[lst,IntegerPart[a]],{n,0,4}];lst
nxt[{n_, a_}] := {n + 1, a^n + n^a}; NestList[nxt, {1, 1}, 4][[All, 2]] (* Harvey P. Dale, Jul 18 2021 *)

Edited by N. J. A. Sloane, May 29 2009

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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A051442 a(n) = n^(n+1)+(n+1)^n.

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A173054 Numbers of the form x^y + y^x, 1 < x < y.

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A055652 Table T(m,k)=m^k+k^m (with 0^0 taken to be 1) as square array read by antidiagonals.

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A045575 Nonnegative numbers of the form x^y - y^x, for x,y > 1.

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A155539 a(n) = n^(n+3) + (n+3)^n.

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A173055 Numbers of the form a^b+b^a, a and b are odd primes, b > a.

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A386892 Numbers k expressible as x^y + y^z + z^x, where x, y, and z are integers > 1.

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