A001585 -id:A001585 - cp's OEIS frontend

A001589 a(n) = 4^n + n^4.

1, 5, 32, 145, 512, 1649, 5392, 18785, 69632, 268705, 1058576, 4208945, 16797952, 67137425, 268473872, 1073792449, 4295032832, 17179952705, 68719581712, 274878037265, 1099511787776, 4398046705585, 17592186278672

Offset: 0

N. J. A. Sloane

a(n) is prime if and only if n = 1. - Reinhard Zumkeller, May 24 2009

The statement above (and the corollary that 5 is the only prime term in this sequence) can be proved using Sophie Germain's identity x^4 + 4y^4 = (x^2 + 2xy + 2y^2)(x^2 - 2xy + 2y^2). - Alonso del Arte, Oct 31 2013 [exponents corrected by Thomas Ordowski, Nov 29 2017]

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Notes, Mathematical Intelligencer 2(2) (1980), p. 66. - _Reinhard Zumkeller_, May 24 2009
Index entries for linear recurrences with constant coefficients, signature (9,-30,50,-45,21,-4).

Cf. A001580, A001585.

[4^n+n^4: n in [0..30]]; // Vincenzo Librandi, Oct 27 2011

Table[4^n + n^4, {n, 0, 40}] (* Vladimir Joseph Stephan Orlovsky, Feb 14 2011 *)
LinearRecurrence[{9,-30,50,-45,21,-4},{1,5,32,145,512,1649},30] (* Harvey P. Dale, Mar 06 2023 *)

PARI
```
a(n)=1<<(n+n)+n^4
```

G.f.: -(5*x^5 + 38*x^4 + 43*x^3 - 17*x^2 + 4*x - 1) / ((x - 1)^5*(4*x - 1)). - Colin Barker, Jan 01 2013

A001593 a(n) = 5^n + n^5.

Original entry on oeis.org

1, 6, 57, 368, 1649, 6250, 23401, 94932, 423393, 2012174, 9865625, 48989176, 244389457, 1221074418, 6104053449, 30518337500, 152588939201, 762940872982, 3814699155193, 19073488804224, 95367434840625, 476837162287226, 2384185796169257, 11920928961514468

Offset: 0

N. J. A. Sloane

a(24) is prime; a(1036) and a(104824) are probable primes (3-PRP). - David Radcliffe, Dec 23 2016

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (11,-45,95,-115,81,-31,5).

Cf. A001580, A001585, A001589.

[5^n+n^5: n in [0..30]]; // Vincenzo Librandi, Oct 27 2011

seq(seq(k^n+n^k, k=5..5), n=0..20); # Zerinvary Lajos, Jun 29 2007

f[n_]:=5^n+n^5;f[Range[0,40]] (* Vladimir Joseph Stephan Orlovsky, Feb 14 2011 *)
LinearRecurrence[{11,-45,95,-115,81,-31,5},{1,6,57,368,1649,6250,23401},30] (* Harvey P. Dale, Jan 15 2018 *)

a(n)=5^n+n^5 \\ Charles R Greathouse IV, Oct 07 2015

G.f.: (4*x^6+135*x^5+289*x^4+84*x^3-36*x^2+5*x-1) / ((x-1)^6*(5*x-1)). - Colin Barker, May 07 2013

More terms from Colin Barker, May 07 2013

A001594 a(n) = 6^n + n^6.

Original entry on oeis.org

1, 7, 100, 945, 5392, 23401, 93312, 397585, 1941760, 10609137, 61466176, 364568617, 2179768320, 13065520825, 78371693632, 470196375201, 2821126684672, 16926683582305, 101559990680640, 609359787056377

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (13,-63,161,-245,231,-133,43,-6).

Cf. sequences of the form k^n+n^k: A001580 (k=2), A001585 (k=3), A001589 (k=4), A001593 (k=5), this sequence (k=6), A001596 (k=7), A198401 (k=8), A185277 (k=9), A177068 (k=10), A177069 (k=11).

[6^n+n^6: n in [0..30]]; // Vincenzo Librandi, Oct 27 2011

seq(seq(k^n+n^k, k=6..6), n=0..19); # Zerinvary Lajos, Jun 29 2007

Table[6^n + n^6, {n, 0, 30}] (* or *) CoefficientList[Series[(1 - 6 x + 72 x^2 - 75 x^3 - 1475 x^4 - 1776 x^5 - 334 x^6 - 7 x^7)/((1-x)^7 (1-6 x)), {x, 0, 30}], x] (* Vincenzo Librandi, Aug 28 2014 *)
LinearRecurrence[{13,-63,161,-245,231,-133,43,-6},{1,7,100,945,5392,23401,93312,397585},20] (* Harvey P. Dale, Jan 07 2023 *)

a(n)=6^n+n^6 \\ Charles R Greathouse IV, Feb 14 2011

[6^n+n^6 for n in (0..30)] # Bruno Berselli, Aug 28 2014

G.f.: (1 - 6*x + 72*x^2 - 75*x^3 - 1475*x^4 - 1776*x^5 - 334*x^6 - 7*x^7)/((1-x)^7*(1-6*x)). - Vincenzo Librandi, Aug 28 2014

A001596 a(n) = 7^n + n^7.

Original entry on oeis.org

1, 8, 177, 2530, 18785, 94932, 397585, 1647086, 7861953, 45136576, 292475249, 1996813914, 13877119009, 96951758924, 678328486353, 4747732369318, 33233199005057, 232630924325880, 1628414210130481

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (15,-84,252,-462,546,-420,204,-57,7).

Cf. A001580, A001585, A001589, A001593, A001594.

[7^n+n^7: n in [0..30]]; // Vincenzo Librandi, Oct 27 2011

seq(seq(k^n+n^k, k=7..7), n=0..18); # Zerinvary Lajos, Jun 29 2007

f[n_]:=7^n+n^7;f[Range[0,40]] (* Vladimir Joseph Stephan Orlovsky, Feb 14 2011 *)
LinearRecurrence[{15,-84,252,-462,546,-420,204,-57,7},{1,8,177,2530,18785,94932,397585,1647086,7861953},20] (* Harvey P. Dale, Sep 16 2018 *)

PARI
```
a(n)=7^n+n^7
```

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]

A185277 a(n) = n^9 + 9^n.

Original entry on oeis.org

1, 10, 593, 20412, 268705, 2012174, 10609137, 45136576, 177264449, 774840978, 4486784401, 33739007300, 287589316833, 2552470327702, 22897453501745, 205929575454024, 1853088908328577, 16677300287543066, 150094833656289489

Offset: 0

Vladimir Joseph Stephan Orlovsky, Feb 19 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (19,-135,525,-1290,2142,-2478,2010,-1125,415,-91,9).

Cf. sequences of the form k^n+n^k: A001580 (k=2), A001585 (k=3), A001589 (k=4), A001593 (k=5), A001594 (k=6), A001596 (k=7), A198401 (k=8), this sequence (k=9), A177068 (k=10), A177069 (k=11).

[9^n+n^9: n in [0..30]]; // Vincenzo Librandi, Oct 27 2011

Table[9^n + n^9, {n, 0, 30}] (* or *) CoefficientList[Series[(1 - 9 x + 538 x^2 + 9970 x^3 - 43028 x^4 - 638168 x^5 - 1317266 x^6 - 779618 x^7 - 130925 x^8 - 4527 x^9 - 8 x^10)/((1 - x)^10 (1 - 9 x)), {x, 0, 30}], x] (* Vincenzo Librandi, Aug 28 2014 *)
LinearRecurrence[{19,-135,525,-1290,2142,-2478,2010,-1125,415,-91,9},{1,10,593,20412,268705,2012174,10609137,45136576,177264449,774840978,4486784401},20] (* Harvey P. Dale, Jun 08 2023 *)

for(n=0,25, print1(n^9 + 9^n, ", ")) \\ G. C. Greubel, Jun 25 2017

[9^n+n^9 for n in (0..30)] # Bruno Berselli, Aug 28 2014

G.f.: (1 - 9*x + 538*x^2 + 9970*x^3 - 43028*x^4 - 638168*x^5-1317266*x^6 - 779618*x^7 - 130925*x^8 - 4527*x^9 - 8*x^10)/((1-x)^10*(1-9*x)). - Vincenzo Librandi, Aug 28 2014

A177069 11^n + n^11.

Original entry on oeis.org

1, 12, 2169, 178478, 4208945, 48989176, 364568617, 1996813914, 8804293473, 33739007300, 125937424601, 570623341222, 3881436747409, 36314872537968, 383799398752905, 4185897925275026, 45967322049616577, 505481300395601404

Offset: 0

Vincenzo Librandi, May 31 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..950
Index entries for linear recurrences with constant coefficients, signature (23,-198,946,-2915,6237,-9636,10956,-9207,5665,-2486,738,-133,11).

Cf. sequences of the form k^n+n^k: A001580 (k=2), A001585 (k=3), A001589 (k=4), A001593 (k=5), A001594 (k=6), A001596 (k=7), A198401 (k=8), A185277 (k=9), A177068 (k=10), this sequence (k=11).

Magma
```
[11^n+n^11: n in [0..20]]
```

Table[11^n + n^11, {n, 0, 30}] (* or *) CoefficientList[Series[(1 - 11 x + 2091 x^2 + 130021 x^3 + 524976 x^4 -14501046 x^5 - 91394082 x^6 - 163229406 x^7 - 104915271 x^8 - 24085951 x^9 - 1676905 x^10 - 22407 x^11 - 10 x^12)/((1 - x)^12 (1 - 11 x)), {x, 0, 30}], x] (* Vincenzo Librandi, Aug 28 2014 *)

a(n)= 11^n+n^11 \\ Charles R Greathouse IV, Jan 11 2012

[11^n+n^11 for n in (0..30)] # Bruno Berselli, Aug 28 2014

G.f.: (1 - 11*x + 2091*x^2 + 130021*x^3 + 524976*x^4 - 14501046*x^5 - 91394082*x^6 - 163229406*x^7 - 104915271*x^8 - 24085951*x^9 - 1676905*x^10 - 22407*x^11 - 10*x^12) / ((1-x)^12*(1-11*x)). - Vincenzo Librandi, Aug 28 2014

A198401 a(n) = 8^n + n^8.

Original entry on oeis.org

1, 9, 320, 7073, 69632, 423393, 1941760, 7861953, 33554432, 177264449, 1173741824, 8804293473, 69149458432, 550571544609, 4399522300160, 35186934979457, 281479271677952, 2251806789442689, 18014409529442560, 144115205059418913

Offset: 0

Vincenzo Librandi, Oct 27 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (17,-108,372,-798,1134,-1092,708,-297,73,-8).

Cf. A001580, A001585, A001589, A001593, A001594, A001596, A177068, A177069, A185277.

Magma
```
[8^n+n^8: n in [0..20]]
```

f[n_] := 8^n + n^8; f[Range[0, 30]]
LinearRecurrence[{17,-108,372,-798,1134,-1092,708,-297,73,-8},{1,9,320,7073,69632,423393,1941760,7861953,33554432,177264449},30] (* Harvey P. Dale, Aug 26 2023 *)

a(n)=8^n+n^8 \\ Charles R Greathouse IV, Oct 16 2015

G.f.: -(9*x^9 +1966*x^8 +34133*x^7 +120575*x^6 +109459*x^5 +18599*x^4 -2233*x^3 -275*x^2 +8*x -1) / ((x -1)^9*(8*x -1)). - Colin Barker, Sep 13 2013

A109273 Numbers k such that k+1 is the smallest prime factor of k^3 + 3^k.

Original entry on oeis.org

1, 4, 10, 16, 22, 28, 40, 52, 58, 70, 88, 100, 112, 130, 136, 148, 196, 232, 238, 250, 256, 280, 310, 316, 352, 382, 400, 418, 430, 442, 478, 490, 556, 562, 592, 598, 616, 640, 682, 742, 772, 796, 808, 820, 862, 880, 928, 970, 976, 1030, 1048, 1060, 1096, 1102

Offset: 1

Zak Seidov and Max Alekseyev, Jun 25 2005

nonn

Except for the first term, 2k+1 is composite.

Cf. A001585.

isok(n) = factor(n^3+3^n)[1, 1] == (n+1); \\ Michel Marcus, Oct 06 2013

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

cp's OEIS Frontend

A001589 a(n) = 4^n + n^4.

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A001593 a(n) = 5^n + n^5.

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A001594 a(n) = 6^n + n^6.

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A001596 a(n) = 7^n + n^7.

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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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A185277 a(n) = n^9 + 9^n.

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A177069 11^n + n^11.

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Sage