A001589 -id:A001589 - cp's OEIS frontend

A089485 Numbers k such that k^4 + 4^k = A001589(k) is a semiprime.

3, 5, 15, 35, 55

Offset: 1

Hugo Pfoertner, Nov 11 2003

For n = 2*k + 1, n^4 + 4^n = (n^2 + n*2^(k + 1) + 2^n) * (n^2 - n*2^(k + 1) + 2^n) The sequence gives those values of n for which both parentheses are primes. No further terms were found for k<=5000.

a(6) > 120000, if it exists. - Tyler Busby, Feb 13 2023

			a(1)=3 because 3^4+4^3=145=5*29, a(2)=5 because 5^4+4^5=1649=17*97.

Ignacio Larrosa Cañestro et al., Find all primes of the form 4^n + n^4. Discussion in newsgroup sci.math (2003).

Cf. A001589.

Select[Range[60],PrimeOmega[#^4+4^#]==2&] (* Harvey P. Dale, Jul 31 2020 *)

for(k=0,5000,my(n=2*k+1,p1=n^2+n*2^(k+1)+2^n,p2=n^2-n*2^(k+1)+2^n);if(ispseudoprime(p1)&&ispseudoprime(p2),print1(n,", "))) \\ Hugo Pfoertner, Jul 24 2019

A045572 Numbers that are odd but not divisible by 5.

Original entry on oeis.org

1, 3, 7, 9, 11, 13, 17, 19, 21, 23, 27, 29, 31, 33, 37, 39, 41, 43, 47, 49, 51, 53, 57, 59, 61, 63, 67, 69, 71, 73, 77, 79, 81, 83, 87, 89, 91, 93, 97, 99, 101, 103, 107, 109, 111, 113, 117, 119, 121, 123, 127, 129, 131, 133, 137, 139, 141, 143, 147, 149, 151, 153

Offset: 1

Jeff Burch, Dec 11 1999

Contains the repunits R_n, (A000042 or A002275): For any m in the sequence (divisible by neither 2 nor 5), Euler's theorem (i.e., m | 10^m - 1 = 9*R_n) guarantees that R_n is always some multiple of m (see A099679) and thus forms a subsequence. - Lekraj Beedassy, Oct 26 2004
Inverse formula: n = 4*floor(a(n)/10) + floor((a(n) mod 10)/3) + 1. - Carl R. White, Feb 06 2008
Numbers ending with 1, 3, 7 or 9. - Lekraj Beedassy, Apr 04 2009
Complement of A065502. - Reinhard Zumkeller, Nov 15 2009
Union of evenish and oddish numbers, cf. A045797, A045798. - Reinhard Zumkeller, Dec 10 2011
Numbers k such that k^(4*j) mod 10 = 1, for any j. - Gary Detlefs, Jan 03 2012
Numbers coprime to 10. - Charles R Greathouse IV, Sep 05 2013
This is also the sequence of numbers such that all their divisors are the sum of the proper divisors of some number (see A001065 (sum of proper divisors) and A078923 (possible values of sigma(n)-n)). This is due to the fact that in the set of untouchable numbers (A005114) there are only 2 prime numbers (2 and 5) and all other terms are even composite. - Michel Marcus, Jun 14 2014
Numbers n for which A001589(n) is divisible by 5. - Bruno Berselli, Jun 18 2014
For a(n) > 1, positive integers x such that the decimal representation of 1/x is purely periodic after the decimal point (1/x is a repeating decimal with no non-repeating portion). - Doug Bell, Aug 05 2015
The asymptotic density of this sequence is 2/5. - Amiram Eldar, Oct 18 2020

			a(18) = 10*floor(17/4) + 2*floor( (4*(17 mod 4) + 1)/3 ) + 1
      = 10*4 + 2*floor( (4*(1)+1)/3 ) + 1
      = 40 + 2*floor(5/3) + 1
      = 40 + 2*1 + 1
      = 43.
G.f. = x + 3*x^2 + 7*x^3 + 9*x^4 + 11*x^5 + 13*x^6 + 17*x^7 + 19*x^8 + ...

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Relative complement of A017329 in A005408.

Cf. A000035, A000042, A001065, A001589, A002275, A005114, A045797, A045798, A065502, A078923, A079998, A082768 (numbers that begin with 1, 3, 7 or 9), A085820, A099679.

a045572 n = a045572_list !! (n-1)
a045572_list = filter ((/= 0) . (`mod` 5)) a005408_list
-- Reinhard Zumkeller, Dec 10 2011

[ 2*n + 2*Floor((n-3)/4) + 1: n in [1..70] ]; // Vincenzo Librandi, Aug 01 2011

A045572:=n->2*n + 2*floor((n-3)/4) + 1; seq(A045572(n), n=1..50); # Wesley Ivan Hurt, Jun 14 2014

Flatten[Table[10n + {1, 3, 7, 9}, {n, 0, 19}]] (* Alonso del Arte, Jan 13 2012 *)
Select[2Range@78 -1, Mod[#, 5] > 0 &] (* Robert G. Wilson v, Apr 02 2017 *)
Map[(1/2*(5*# + Mod[3*# + 2, 4] - 4))&, Range[10^3]] (* Mikk Heidemaa, Nov 23 2017 *)

a(n)=10*(n>>2)+[-1,1,3,7][n%4+1] \\ Charles R Greathouse IV, Jul 31 2011

is(n)=gcd(n,10)==1 \\ Charles R Greathouse IV, Sep 05 2013

{a(n) = 2*n - 1 + (n+1) \ 4 * 2}; /* Michael Somos, Jun 15 2014 */

def A045572(n): return 2*(n+(n+1)//4) - 1 # Chai Wah Wu, Jan 08 2022

a(n) = 10*floor((n-1)/4) + 2*floor( (4*((n-1) mod 4) + 1)/3 ) + 1; a(n) = a(n-1) + 2 + 2*floor(((x+6) mod 10)/9). - Carl R. White, Feb 06 2008
a(n) = 2*n + 2*floor((n-3)/4) + 1. - Kenneth Hammond (weregoose(AT)gmail.com), Mar 07 2008
a(n) = -1 + 2*n + 2*floor((n+1)/4). - Kenneth Hammond (weregoose(AT)gmail.com), Mar 25 2008
From R. J. Mathar, Sep 22 2009: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5).
G.f.: x*(1 + 2*x + 4*x^2 + 2*x^3 + x^4)/((1+x) * (x^2+1) * (x-1)^2). (End)
A000035(a(n))*(1 - A079998(a(n))) = 1. - Reinhard Zumkeller, Nov 15 2009
a(n) = (10*n + 2*(-1)^(n*(n+1)/2) - (-1)^n - 5)/4. - Bruno Berselli, Nov 06 2011
G.f.: x * (1 + 2*x + 4*x^2 + 2*x^3 + x^4) / ((1 - x) * (1 - x^4)). - Michael Somos, Jun 15 2014
a(1 - n) = -a(n) for all n in Z. - Michael Somos, Jun 15 2014
0 = (a(n) - 2*a(n+1) + a(n+2)) * (a(n) - 4*a(n+2) + 3*a(n+3)) for all n in Z. - Michael Somos, Jun 15 2014
From Mikk Heidemaa, Nov 22 2017: (Start)
a(n) = (1/2)*(5*n + ((3*n + 2) mod 4) - 4);
a(n) = (1/4)*((-1)^(n + 1) + 10*n + 2*cos((n*Pi)/2) - 2*sin((n*Pi)/2) - 5);
a(n) = (1/4)*((-1)^(1 + n) + (1 - i)*exp(-(1/2)*i*n*Pi) + (1 + i)*exp(i*n*Pi/2) + 10*n - 5) (for n > 0), where i is the imaginary unit. (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt(10-2*sqrt(5))*Pi/10. - Amiram Eldar, Dec 12 2021
E.g.f.: (2 + cos(x) + (5*x - 3)*cosh(x) - sin(x) + (5*x - 2)*sinh(x))/2. - Stefano Spezia, Dec 07 2022

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

A349265 Generalized Euler numbers, a(n) = n![x^n](sec(5x)(sin(x) + sin(3x) + cos(2x) + cos(4x))).

Original entry on oeis.org

2, 4, 30, 272, 3522, 55744, 1066590, 23750912, 604935042, 17328937984, 551609685150, 19313964388352, 737740947722562, 30527905292468224, 1360427147514751710, 64955605537174126592, 3308161927353377294082, 179013508069217017790464, 10256718523496425979562270

Offset: 0

Peter Luschny, Nov 20 2021

nonn

For references and examples see A349264.

Matthew House, Table of n, a(n) for n = 0..377

Row 5 of A349271.

Cf. A000111, A001589, A007289, A349264, A001587.

m = 18; CoefficientList[Series[Sec[5*x] * (Sin[x] + Sin[3*x] + Cos[2*x] + Cos[4*x]), {x, 0, m}], x] * Range[0, m]! (* Amiram Eldar, Nov 20 2021 *)

seq(n)={my(x='x + O('x^(n+1))); Vec(serlaplace((sin(x) + sin(3*x) + cos(2*x) + cos(4*x))/cos(5*x)))} \\ Andrew Howroyd, Nov 20 2021

t = PowerSeriesRing(QQ, 't', default_prec=19).gen()
f = (sin(t) + sin(3*t) + cos(2*t) + cos(4*t)) / cos(5*t)
f.egf_to_ogf().list()

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

cp's OEIS Frontend

A089485 Numbers k such that k^4 + 4^k = A001589(k) is a semiprime.

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A045572 Numbers that are odd but not divisible by 5.

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Python

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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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Formula

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

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A349265 Generalized Euler numbers, a(n) = n![x^n](sec(5x)(sin(x) + sin(3x) + cos(2x) + cos(4x))).