A074606 -id:A074606 - cp's OEIS frontend

A122853 Numbers k such that (3^k + 5^k)/8 = A074606(k)/8 is a prime.

3, 5, 7, 17, 19, 109, 509, 661, 709, 1231, 12889, 13043, 26723, 43963, 44789

Offset: 1

Alexander Adamchuk, Sep 14 2006

(3^k + 5^k)/8 = A074606(k)/8 = A081186(k)/4.
Corresponding primes of the form (3^k + 5^k)/2^3 are listed in {A121938(n)} = {A079773(a(n))} = {19, 421, 10039, 95383574161, 2384331073699, ...}.
No other terms less than 100000. - Robert Price, Apr 28 2012

Cf. A074606, A081186, A121824, A121877, A005058, A005059, A121938, A109347, A079773.

Do[f=5^n+3^n;If[PrimeQ[f/2^3],Print[{n,f/2^3}]],{n,1,1231}]

select(n->isprime((3^n + 5^n)/8), vector(2000,i,i)) \\ Charles R Greathouse IV, Feb 13 2011

a(11)-a(15) from Robert Price, Apr 28 2012

A121938 Primes of the form (3^k + 5^k)/2^3 = A074606(k)/8.

Original entry on oeis.org

19, 421, 10039, 95383574161, 2384331073699, 1925929944387235853055979210606894889560480247048440342330377620014353281101

Offset: 1

Zak Seidov, Sep 10 2006

Corresponding numbers k such that (3^k + 5^k)/8 is prime are listed in A122853. All these numbers are primes. - Alexander Adamchuk, Sep 14 2006

The next term is too large to include. - Alexander Adamchuk, Sep 14 2006

Cf. A074706, A122853, A081186, A079773, A121824, A121877, A005058, A005059, A121938, A109347.

Do[f=5^n+3^n;If[PrimeQ[f/2^3],Print[{n,f/2^3}]],{n,1,1231}] (* Alexander Adamchuk, Sep 14 2006 *)

a(n) = (A122853(n)^3 + A122853(n)^5)/8. a(n) = A074606[A122853(n)]/8 = A081186[A122853(n)]/4. a(n) = A079773[A122853(n)]. - Alexander Adamchuk, Sep 14 2006

More terms from Alexander Adamchuk, Sep 14 2006

A155603 a(n) = 5^n+3^n-1.

Original entry on oeis.org

1, 7, 33, 151, 705, 3367, 16353, 80311, 397185, 1972807, 9824673, 49005271, 244672065, 1222297447, 6108298593, 30531927031, 152630937345, 763068593287, 3815084686113, 19074648589591, 95370918425025, 476847618556327

Offset: 0

Mohammad K. Azarian, Jan 25 2009

Index entries for linear recurrences with constant coefficients, signature (9,-23,15).

Cf. A074501, A020515, A155588, A155590, A155592, A155593, A155594, A155596, A155597, A155598, A155599, A155600, A155601, A155602.

Table[5^n+3^n-1,{n,0,30}] (* or *) LinearRecurrence[{9,-23,15},{1,7,33},30] (* Harvey P. Dale, Oct 14 2020 *)

a(n)=5^n+3^n-1 \\ Charles R Greathouse IV, Jun 11 2015

G.f.: 1/(1-5*x)+1/(1-3*x)-1/(1-x).
E.g.f.: e^(5*x)+e^(3*x)-e^x.
a(n) = 8*a(n-1)-15*a(n-2)-8 with a(0)=1, a(1)=7. - Vincenzo Librandi, Jul 21 2010
a(n) = A074606(n)-1. - R. J. Mathar, Mar 10 2022

A335095 Square array T(n,k), n>=0, k>=0, read by antidiagonals: T(n,k) = ((2n+1)!!)^k * Sum_{j=1..n} 1/(2*j+1)^k.

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 0, 1, 8, 3, 0, 1, 34, 71, 4, 0, 1, 152, 1891, 744, 5, 0, 1, 706, 55511, 164196, 9129, 6, 0, 1, 3368, 1745731, 41625144, 20760741, 129072, 7, 0, 1, 16354, 57365351, 11575291716, 56246975289, 3616621254, 2071215, 8

Offset: 0

Seiichi Manyama, Sep 12 2020

			Square array begins:
  0,   0,      0,        0,           0, ...
  1,   1,      1,        1,           1, ...
  2,   8,     34,      152,         706, ...
  3,  71,   1891,    55511,     1745731, ...
  4, 744, 164196, 41625144, 11575291716, ...

Column k=0..4 give A001477, A334670, A335090, A335091, A335092.
Rows n=0-2 give: A000004, A000012, A074606.
Main diagonal gives A335096.
Cf. A291656.

T[n_, k_] := ((2*n + 1)!!)^k * Sum[1/(2*j + 1)^k, {j, 1, n}]; Table[T[k, n - k], {n, 0, 8}, {k, 0, n}] // Flatten (* Amiram Eldar, Apr 29 2021 *)

{T(n, k) = prod(j=1, n, 2*j+1)^k*sum(j=1, n, 1/(2*j+1)^k)}

T(0,k) = 0, T(1,k) = 1 and T(n,k) = ((2*n-1)^k+(2*n+1)^k) * T(n-1,k) - (2*n-1)^(2*k) * T(n-2, k) for n>1.

A121824 Primes of the form (3^n + 5^n)/2.

Original entry on oeis.org

17, 353, 198593

Offset: 1

Zak Seidov, Aug 27 2006

Corresponding n are 2^1, 2^2, 2^3. What are the following terms? Cf. A074606 3^n + 5^n.
Since x^n + y^n has x+y as a factor if n is odd, we can assume that n is a power of 2. Maple shows that up to n = 2^15, there are no more primes of the form (3^n + 5^n)/2. This raises the question: Is it true that x^n + (x+2)^n is irreducible over Q for n a power of 2? - W. Edwin Clark, Sep 10 2006
Next term, if it exists, is > (3^2500+5^2500)/2. - Hugo Pfoertner, Sep 10 2006
No more terms <= (3^(2^17)+5^(2^17))/2=(3^131072+5^131072)/2. Hence the next term, if it exists, is greater than 10^91616 (so is too large to include). - Herman Jamke (hermanjamke(AT)fastmail.fm), Dec 31 2007

Cf. A074606, A121710.

Select[Table[(3^n + 5^n)/2,{n,100}],PrimeQ] (* James C. McMahon, Oct 22 2024 *)

for(n=1,17, m=(3^(2^n)+5^(2^n))/2;if(isprime(m),print1(m","))) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), Dec 31 2007

Edited by N. J. A. Sloane, Jan 13 2008

A245806 a(n) = 3^n + 10^n.

Original entry on oeis.org

2, 13, 109, 1027, 10081, 100243, 1000729, 10002187, 100006561, 1000019683, 10000059049, 100000177147, 1000000531441, 10000001594323, 100000004782969, 1000000014348907, 10000000043046721, 100000000129140163, 1000000000387420489, 10000000001162261467

Offset: 0

Vincenzo Librandi, Aug 04 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (13,-30).

Cf. 3^n+k^n: A034472 (k=1), A007689 (k=2), A008776 (k=3), A074605 (k=4), A074606 (k=5), A074607 (k=6), A074608 (k=7), A074609 (k=8), A074610 (k=9), this sequence (k=10).

Magma
```
[3^n+10^n: n in [0..25]];
```

I:=[2,13]; [n le 2 select I[n] else 13*Self(n-1)-30*Self(n-2): n in [1..25]];

Table[(3^n + 10^n), {n, 0, 30}] (* or *) CoefficientList[Series[(2 - 13 x)/((1 - 3 x) (1 - 10 x)), {x, 0, 30}], x]

a(n)=3^n + 10^n \\ Charles R Greathouse IV, Aug 26 2014

G.f.: (2-13*x)/((1-3*x)(1-10*x)).
E.g.f.: e^(3*x) + e^(10*x).
a(n) = 13*a(n-1)-30*a(n-2) for n>1.
a(n) = A000244(n) + A011557(n). - Michel Marcus, Aug 04 2014

A120948 8n+3^n+5^n.

Original entry on oeis.org

2, 16, 50, 176, 738, 3408, 16402, 80368, 397250, 1972880, 9824754, 49005360, 244672162, 1222297552, 6108298706, 30531927152, 152630937474, 763068593424, 3815084686258, 19074648589744, 95370918425186, 476847618556496

Offset: 0

Mohammad K. Azarian, Aug 19 2006

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,-32,38,-15).

Cf. A074606, A005058.

[8*n+3^n+5^n: n in [0..25]]; // Bruno Berselli, Feb 27 2013

CoefficientList[Series[2 (1 - 2 x - 23 x^2 + 56 x^3)/((1-5 x) (1 - 3 x) (1 - x)^2), {x, 0, 30}], x] (* Vincenzo Librandi, Feb 26 2013 *)
Table[8 n + 3^n + 5^n, {n, 0, 25}] (* Bruno Berselli, Feb 27 2013 *)

for(n=0, 25, print1(8*n+3^n+5^n", ")); \\ Bruno Berselli, Feb 27 2013

G.f.: 2*(1-2*x-23*x^2+56*x^3)/((1-5*x)*(1-3*x)*(1-x)^2). - Vincenzo Librandi, Feb 26 2013

Edited by Ray Chandler, Sep 06 2006

A120949 2n+3^n+5^n.

Original entry on oeis.org

2, 10, 38, 158, 714, 3378, 16366, 80326, 397202, 1972826, 9824694, 49005294, 244672090, 1222297474, 6108298622, 30531927062, 152630937378, 763068593322, 3815084686150, 19074648589630, 95370918425066, 476847618556370

Offset: 0

Mohammad K. Azarian, Aug 19 2006

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,-32,38,-15).

Cf. A074606, A005058.

[2*n+3^n+5^n: n in [0..30]]; // Vincenzo Librandi, Feb 27 2013

Table[2 n + 3^n + 5^n, {n, 0, 30}] (* or *) CoefficientList[Series[2 (1 - 5 x + x^2 + 11 x^3)/((1 -5 x) (1 - 3 x) (1 - x)^2), {x, 0, 30}], x] (* Vincenzo Librandi, Feb 26 2013 *)
LinearRecurrence[{10,-32,38,-15},{2,10,38,158},30] (* Harvey P. Dale, Jan 18 2021 *)

a(n)=2*n+3^n+5^n \\ Charles R Greathouse IV, Feb 27 2013

G.f.: 2*(1-5*x+x^2+11*x^3)/((1-5*x)*(1-3*x)*(1-x)^2). - Vincenzo Librandi, Feb 26 2013

Edited by Ray Chandler, Sep 06 2006

A120950 3^n+5^n-2n.

Original entry on oeis.org

2, 6, 30, 146, 698, 3358, 16342, 80298, 397170, 1972790, 9824654, 49005250, 244672042, 1222297422, 6108298566, 30531927002, 152630937314, 763068593254, 3815084686078, 19074648589554, 95370918424986, 476847618556286

Offset: 0

Mohammad K. Azarian, Aug 19 2006

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,-32,38,-15).

Cf. A074606, A005058.

[3^n+5^n-2*n: n in [0..30]]; // Vincenzo Librandi, Feb 27 2013

Table[3^n + 5^n - 2 n, {n, 0, 30}] (* or *) CoefficientList[Series[2 (1 - 7 x + 17 x^2 - 19 x^3)/((1 - 5 x) (1 - 3 x) (1 - x)^2), {x, 0, 30}], x] (* Vincenzo Librandi, Feb 26 2013 *)
LinearRecurrence[{10,-32,38,-15},{2,6,30,146},30] (* Harvey P. Dale, Apr 27 2016 *)

G.f.: 2*(1-7*x+17*x^2-19*x^3)/((1-5*x)*(1-3*x)*(1-x)^2). - Vincenzo Librandi, Feb 26 2013

Edited by Ray Chandler, Sep 06 2006

A120969 a(n) = 8n+5^n-3^n.

Original entry on oeis.org

0, 10, 32, 122, 576, 2922, 14944, 75994, 384128, 1933514, 9706656, 48651066, 243609280, 1219108906, 6098732768, 30503229338, 152544844032, 762810313098, 3814309845280, 19072324066810, 95363944856384, 476826697850090

Offset: 0

Mohammad K. Azarian, Aug 19 2006

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,-32,38,-15).

Cf. A074606, A005058.

[8*n+5^n-3^n: n in [0..30]]; // Vincenzo Librandi, Feb 25 2013

I:=[0,10,32,122]; [n le 4 select I[n] else 10*Self(n-1)-32*Self(n-2)+38*Self(n-3)-15*Self(n-4): n in [1..30]]; // Vincenzo Librandi, Feb 25 2013

CoefficientList[Series[2 x (5 - 34 x + 61 x^2)/((1-x)^2(1-5x)(1-3x)), {x, 0, 30}], x] (* Vincenzo Librandi, Feb 23 2013 *)

G.f.: 2*x*(5-34*x+61*x^2)/((1-x)^2*(1-5*x)*(1-3*x)). - Vincenzo Librandi, Feb 23 2013

Edited by Ray Chandler, Sep 06 2006

cp's OEIS Frontend

A122853 Numbers k such that (3^k + 5^k)/8 = A074606(k)/8 is a prime.

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A121938 Primes of the form (3^k + 5^k)/2^3 = A074606(k)/8.

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A155603 a(n) = 5^n+3^n-1.

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A335095 Square array T(n,k), n>=0, k>=0, read by antidiagonals: T(n,k) = ((2n+1)!!)^k * Sum_{j=1..n} 1/(2*j+1)^k.

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A121824 Primes of the form (3^n + 5^n)/2.

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A245806 a(n) = 3^n + 10^n.

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A120948 8n+3^n+5^n.

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A120949 2n+3^n+5^n.

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