A076980 -id:A076980 - cp's OEIS frontend

A173058 Leyland numbers (Cubes), a^b+b^a, a and b > 1.

8, 512, 1056589062271330492704679569833033213037694652072243044255921418053347805113449718948834511775314375789348789986514257357764695119005371074501077956925879153816773367998010168337463035352852882106048465816422376808296056585503123477676793797534072952979077161795475996672

Offset: 1

Vladimir Joseph Stephan Orlovsky, Feb 08 2010

			2^3=8, 8^3=512,
101851798816724304313422284420468908052573419683296812531807022467719064988166\
8353091698688^3=1056...6672

Cf. A076980, A173054, A173055, A173056

f[a_,b_]:=a^b+b^a; Select[Union[Flatten[Table[f[a,b],{a,2,150},{b,2,150}]]],IntegerQ[(#1)^(1/3)]&]

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)
```

A284646 Variation on Leyland numbers: k = x'^y + y'^x, where x' and y' are the arithmetic derivative of x and y.

Original entry on oeis.org

2, 17, 26, 37, 50, 65, 82, 101, 126, 145, 170, 197, 217, 226, 257, 325, 344, 362, 401, 442, 485, 512, 513, 577, 626, 677, 730, 785, 901, 962, 1001, 1025, 1090, 1157, 1297, 1445, 1522, 1601, 1682, 1729, 1765, 1850, 1937, 2026, 2117, 2198, 2305, 2402, 2501, 2602

Offset: 1

Paolo P. Lava, Mar 31 2017

nonn

Another similar variation on Leyland numbers is k = x^y' + y^x' that leads to A014091.

			2' = 1, 4' = 4, 1^4 + 4^2 = 1 + 16 = 17.

Cf. A014091, A076980.

with(numtheory): N:= 10^5: A:={}: for x from 2 to floor(N^(1/2)) do
for y from 2 do yd:=y*add(op(2,p)/op(1,p),p=ifactors(y)[2]); xd:=x*add(op(2,p)/op(1,p),p=ifactors(x)[2]); a:= xd^y + yd^x;
if a>N then break fi; A:=A union {a}; od; od; sort([op(A)]);
# based on Robert Israel code in A076980.

A321518 Smallest k > 1 such that n^k + k^n is prime, i.e., a Leyland prime, or 0 if no such k exists.

Original entry on oeis.org

3, 2, 0, 24

Offset: 2

Felix Fröhlich, Nov 12 2018

a(4) = 0. Proof: For k == 1 (mod 4), 4^k + k^4 = 4*x^4 + k^4 = (2*x^2 - 2*k*x + k^2)(2*x^2 + 2*k*x + k^2), where x = 4^((k-1)/4). For k == 3 (mod 4), 4^k + k^4 = 64*x^4 + k^4 = (8*x^2 - 4*k*x + k^2)(8*x^2 + 4*k*x + k^2), where x = 4^((k-3)/4) (cf. Israel, 2015).
Conjecture: a(6) = 0.
From Jon E. Schoenfield, Nov 13 2018: (Start)
Let t = 6^k + k^6.
If k is even, then 2|t.
If k is odd but not divisible by 7, then 7|t.
If k is divisible by 3, then 3|t.
If k == 7 or 63 (mod 70), then 5|t.
Thus, a(6) == 35, 49, 91, 119, 161, or 175 (mod 210) if a(6) > 0. (End)

			For n = 5: 5^24 + 24^5 = 59604644783353249 is prime, and 24 is the smallest k > 1 such that 5^k + k^5 is prime, so a(5) = 24.

Robert Israel, Re: Smallest k > 1 such that n^k+k^n is prime, or 0 if no such k exists, Seqfan (Dec 11 2015).
Wikipedia, Leyland number

Cf. A076980, A094133.

A356423 Leyland numbers which are products of two distinct primes.

Original entry on oeis.org

57, 145, 177, 1649, 7073, 23401, 131361, 423393, 2012174, 4785713, 33555057, 43050817, 177264449, 364568617, 1073792449, 4486784401, 13877119009, 31381070257, 94143190994, 125937424601, 2552470327702, 8796093024057, 33233199005057, 130291290501553, 1628414210130481, 1853020188884609

Offset: 1

Massimo Kofler, Aug 07 2022

nonn

A squarefree subsequence of Leyland numbers (which are numbers that can be written as a^b + b^a for a, b > 1).

			57 = 3*19 = 5^2 + 2^5.
2012174 = 2*1006087 = 9^5 + 5^9.
4486784401 = 11*407889491 = 10^9 + 9^10.
2552470327702 = 2*1276235163851 = 13^9 + 9^13.

Intersection of A076980 and A006881.

A373867 Perfect powers of the form x^y + y^x, where x > 1 and y > 1.

Original entry on oeis.org

8, 32, 100, 512, 33554432, 36893488147419103232, 2923003274661805836407369665432566039311865085952, 78804012392788958424558080200287227610159478540930893335896586808491443542994421222828532509769831281613255980613632

Offset: 1

Gonzalo Martínez, Jun 21 2024

nonn

Subsequence of A076980: a(n) is a Leyland number that is a perfect power. The condition that x > 1 and y > 1 is necessary, otherwise every perfect power would belong to this sequence, since m^n = (m^n-1)^1 + 1^(m^n-1).
If x = y = 2^k, then x^y + y^x = 2^(k*2^k + 1) belongs to this sequence for all k > 0, and (k*2^k + 1) is the k-th Cullen number. That is, 2^A002064(k) is a term, with k > 0, from which it follows that this sequence has infinitely many terms.
Conjecture: 32 and 100 are the only terms for which x != y: 2^4 + 4^2 = 2^5 = 32 and 2^6 + 6^2 = 10^2 = 100.

			100 is a term, because 100 = 10^2 and 100 = 2^6 + 6^2.

Cf. A001597, A002064, A076980.

cp's OEIS Frontend

A173058 Leyland numbers (Cubes), a^b+b^a, a and b > 1.

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Mathematica

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

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PARI

A284646 Variation on Leyland numbers: k = x'^y + y'^x, where x' and y' are the arithmetic derivative of x and y.

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Maple

A321518 Smallest k > 1 such that n^k + k^n is prime, i.e., a Leyland prime, or 0 if no such k exists.

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A356423 Leyland numbers which are products of two distinct primes.

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A373867 Perfect powers of the form x^y + y^x, where x > 1 and y > 1.

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