A158879 -id:A158879 - cp's OEIS frontend

A104745 a(n) = 5^n + n.

1, 6, 27, 128, 629, 3130, 15631, 78132, 390633, 1953134, 9765635, 48828136, 244140637, 1220703138, 6103515639, 30517578140, 152587890641, 762939453142, 3814697265643, 19073486328144, 95367431640645, 476837158203146, 2384185791015647, 11920928955078148, 59604644775390649

Offset: 0

Zak Seidov, Mar 23 2005

Numbers m=5^n+n such that equation x=5^(m-x) has solution x=5^n, see A104744.
No primes of the form 5^n+n for n < 7954. - Thomas Ordowski, Oct 28 2013
a(7954) is prime (5560 digits). - Thomas Ordowski, May 07 2015

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Factordb, 5^7954 + 7954.
Index entries for linear recurrences with constant coefficients, signature (7,-11,5).

Cf. A006127, A058046, A081552, A093324, A104743, A104744, A158879, A181572.

I:=[1, 6, 27]; [n le 3 select I[n] else 7*Self(n-1)-11*Self(n-2) +5*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Jun 16 2013

g:=1/(1-5*z): gser:=series(g, z=0, 43): seq(coeff(gser, z, n)+n, n=0..31); # Zerinvary Lajos, Jan 09 2009

Table[5^n+n,{n,0,40}] (* Vladimir Joseph Stephan Orlovsky, May 19 2011 *)
CoefficientList[Series[(1 - x - 4 x^2) / ((1 - 5 x) (1 - x)^2), {x, 0, 30}], x] (* Vincenzo Librandi, Jun 16 2013 *)
LinearRecurrence[{7,-11,5},{1,6,27},30] (* Harvey P. Dale, Dec 03 2017 *)

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

From Vincenzo Librandi, Jun 16 2013: (Start)
G.f.: (1-x-4*x^2)/((1-5*x)*(1-x)^2).
a(n) = 7*a(n-1) - 11*a(n-2) + 5*a(n-3). (End)
E.g.f.: exp(x)*(exp(4*x) + x). - Elmo R. Oliveira, Mar 05 2025

More terms from Jonathan R. Love (japanada11(AT)yahoo.ca), Mar 09 2007

A226199 a(n) = 7^n + n.

Original entry on oeis.org

1, 8, 51, 346, 2405, 16812, 117655, 823550, 5764809, 40353616, 282475259, 1977326754, 13841287213, 96889010420, 678223072863, 4747561509958, 33232930569617, 232630513987224, 1628413597910467, 11398895185373162, 79792266297612021, 558545864083284028, 3909821048582988071

Offset: 0

Vincenzo Librandi, Jun 16 2013

Smallest prime of this form is a(34) = 54116956037952111668959660883.

In general, the g.f. of a sequence of numbers of the form k^n + n is (1-x-(k-1)*x^2)/((1-k*x)*(x-1)^2) with main linear recurrence (k+2)*a(n-1) - (2*k+1)*a(n-2) + k*a(n-3). - Bruno Berselli, Jun 16 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Index entries for linear recurrences with constant coefficients, signature (9,-15,7).

Cf. numbers of the form k^n + n: A006127 (k=2), A104743 (k=3), A158879 (k=4), A104745 (k=5), A226200 (k=6), this sequence (k=7), A226201 (k=8), A226202 (k=9), A081552 (k=10), A226737 (k=11).

Cf. A199483 (first differences), A370657.

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

I:=[1, 8, 51]; [n le 3 select I[n] else 9*Self(n-1)-15*Self(n-2)+7*Self(n-3): n in [1..30]];

Table[7^n + n, {n, 0, 30}] (* or *) CoefficientList[Series[(1 - x - 6 x^2) / ((1 - 7 x) (1 - x)^2), {x, 0, 20}], x]
LinearRecurrence[{9,-15,7},{1,8,51},30] (* Harvey P. Dale, Jun 16 2025 *)

a(n)=7^n+n \\ Charles R Greathouse IV, Oct 07 2015

G.f.: (1-x-6*x^2)/((1-7*x)*(1-x)^2).
a(n) = 9*a(n-1) - 15*a(n-2) + 7*a(n-3).
E.g.f.: exp(x)*(exp(6*x) + x). - Elmo R. Oliveira, Mar 05 2025

A226201 a(n) = 8^n + n.

Original entry on oeis.org

1, 9, 66, 515, 4100, 32773, 262150, 2097159, 16777224, 134217737, 1073741834, 8589934603, 68719476748, 549755813901, 4398046511118, 35184372088847, 281474976710672, 2251799813685265, 18014398509482002, 144115188075855891, 1152921504606846996, 9223372036854775829

Offset: 0

Vincenzo Librandi, Jun 16 2013

Smallest prime of this form is a(101). - Bruno Berselli, Jun 17 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Index entries for linear recurrences with constant coefficients, signature (10,-17,8).

Cf. numbers of the form k^n+n: A006127 (k=2), A104743 (k=3), A158879 (k=4), A104745 (k=5), A226200 (k=6), A226199 (k=7), this sequence (k=8), A226202 (k=9), A081552 (k=10), A226737 (k=11).

Cf. A199555 (first differences).

Magma
```
[8^n+n: n in [0..30]];
```

I:=[1, 9, 66]; [n le 3 select I[n] else 10*Self(n-1)-17*Self(n-2)+8*Self(n-3): n in [1..30]];

Table[8^n + n, {n, 0, 30}] (* or *) CoefficientList[Series[(-1 + x + 7 x^2) / ((8 x - 1) (x - 1)^2), {x, 0, 30}], x]
LinearRecurrence[{10,-17,8},{1,9,66},30] (* Harvey P. Dale, Aug 11 2015 *)

a(n)=8^n+n \\ Charles R Greathouse IV, Oct 07 2015

G.f.: (-1+x+7*x^2)/((8*x-1)*(x-1)^2).
a(n) = 10*a(n-1) - 17*a(n-2) + 8*a(n-3).
E.g.f.: exp(x)*(exp(7*x) + x). - Elmo R. Oliveira, Mar 05 2025

A226202 a(n) = 9^n + n.

Original entry on oeis.org

1, 10, 83, 732, 6565, 59054, 531447, 4782976, 43046729, 387420498, 3486784411, 31381059620, 282429536493, 2541865828342, 22876792454975, 205891132094664, 1853020188851857, 16677181699666586, 150094635296999139, 1350851717672992108, 12157665459056928821, 109418989131512359230

Offset: 0

Vincenzo Librandi, Jun 16 2013

After 83, the next prime of this form is a(76). - Bruno Berselli, Jun 18 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Index entries for linear recurrences with constant coefficients, signature (11,-19,9).

Cf. numbers of the form k^n + n: A006127 (k=2), A104743 (k=3), A158879 (k=4), A104745 (k=5), A226200 (k=6), A226199 (k=7), A226201 (k=8), this sequence (k=9), A081552 (k=10), A226737 (k=11).

Cf. A199677 (first differences).

Magma
```
[9^n+n: n in [0..30]];
```

I:=[1, 10, 83]; [n le 3 select I[n] else 11*Self(n-1)-19*Self(n-2)+9*Self(n-3): n in [1..30]];

Table[9^n + n, {n, 0, 30}] (* or *) CoefficientList[Series[(- 1 + x + 8 x^2) / ((9 x - 1) (x - 1)^2), {x, 0, 30}], x]
LinearRecurrence[{11,-19,9},{1,10,83},20] (* Harvey P. Dale, Feb 03 2016 *)

a(n)=9^n+n \\ Charles R Greathouse IV, Oct 07 2015

G.f.: (-1+x+8*x^2)/((9*x-1)*(x-1)^2).
a(n) = 11*a(n-1) - 19*a(n-2) + 9*a(n-3).
E.g.f.: exp(x)*(exp(8*x) + x). - Elmo R. Oliveira, Sep 09 2024

A226200 a(n) = 6^n + n.

Original entry on oeis.org

1, 7, 38, 219, 1300, 7781, 46662, 279943, 1679624, 10077705, 60466186, 362797067, 2176782348, 13060694029, 78364164110, 470184984591, 2821109907472, 16926659444753, 101559956668434, 609359740010515, 3656158440062996, 21936950640377877, 131621703842267158, 789730223053602839

Offset: 0

Vincenzo Librandi, Jun 16 2013

After 7, the next prime of this form has 238 digits (see A058828). - Bruno Berselli, Jun 18 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Index entries for linear recurrences with constant coefficients, signature (8,-13,6).

Cf. numbers of the form k^n + n: A006127 (k=2), A104743 (k=3), A158879 (k=4), A104745 (k=5), this sequence (k=6), A226199 (k=7), A226201 (k=8), A226202 (k=9), A081552 (k=10), A226737 (k=11).

Cf. A058828, A199320 (first differences).

Magma
```
[6^n+n: n in [0..30]];
```

I:=[1, 7, 38]; [n le 3 select I[n] else 8*Self(n-1)-13*Self(n-2)+6*Self(n-3): n in [1..30]];

Table[6^n + n, {n, 0, 30}] (* or *) CoefficientList[Series[(-1 + x + 5 x^2) / ((6 x - 1) (x - 1)^2), {x, 0, 30}], x]

a(n)=6^n+n \\ Charles R Greathouse IV, Oct 07 2015

G.f.: (-1+x+5*x^2)/((6*x-1)*(x-1)^2).
a(n) = 8*a(n-1) - 13*a(n-2) + 6*a(n-3).
E.g.f.: exp(x)*(exp(5*x) + x). - Elmo R. Oliveira, Mar 05 2025

A226737 a(n) = 11^n + n.

Original entry on oeis.org

1, 12, 123, 1334, 14645, 161056, 1771567, 19487178, 214358889, 2357947700, 25937424611, 285311670622, 3138428376733, 34522712143944, 379749833583255, 4177248169415666, 45949729863572177, 505447028499293788, 5559917313492231499, 61159090448414546310, 672749994932560009221

Offset: 0

Vincenzo Librandi, Jun 16 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Index entries for linear recurrences with constant coefficients, signature (13,-23,11).

Cf. numbers of the form k^n + n: A006127 (k=2), A104743 (k=3), A158879 (k=4), A104745 (k=5), A226200 (k=6), A226199 (k=7), A226201 (k=8), A226202 (k=9), A081552 (k=10), this sequence (k=11).

Cf. A199764 (first differences).

Magma
```
[11^n+n: n in [0..30]];
```

I:=[1, 12, 123]; [n le 3 select I[n] else 13*Self(n-1)-23*Self(n-2)+11*Self(n-3): n in [1..30]];

Table[11^n + n, {n, 0, 30}] (* or *) CoefficientList[Series[(- 1 + x + 10 x^2) / ((11 x - 1) (x - 1)^2), {x, 0, 30}], x]
LinearRecurrence[{13,-23,11},{1,12,123},20] (* Harvey P. Dale, Nov 14 2018 *)

a(n)=11^n+n \\ Charles R Greathouse IV, Oct 07 2015

G.f.: (-1+x+10*x^2)/((11*x-1)*(x-1)^2).
a(n) = 13*a(n-1) - 23*a(n-2) + 11*a(n-3).
E.g.f.: exp(x)*(exp(10*x) + x). - Elmo R. Oliveira, Mar 06 2025

A324051 a(n) = A106315(A156552(n)).

Original entry on oeis.org

0, 1, 2, 5, 4, 2, 6, 0, 1, 18, 10, 3, 16, 4, 12, 67, 12, 4, 18, 30, 36, 260, 22, 16, 8, 8, 44, 5, 20, 1029, 30, 28, 164, 36, 28, 6, 256, 96, 44, 4102, 36, 7, 66, 16, 104, 16391, 46, 12, 13, 32, 130, 8, 28, 51, 70, 480, 942, 65544, 42, 9, 2724, 32, 66, 30, 84, 262153, 124, 508, 40, 10, 4, 1048586, 3320, 20, 188, 50, 52, 11, 78, 24

Offset: 2

Antti Karttunen, Feb 19 2019

nonn

Positions of zeros, which is sequence A005940(1+A001599(n)) sorted into ascending order: 2, 9, 125, 325, 351, 4199, ..., has A324201 as its subsequence.

Antti Karttunen, Table of n, a(n) for n = 2..4473
Index entries for sequences computed from indices in prime factorization

Cf. A001599, A005940, A106315, A156552, A158879, A323243, A323244, A324049, A324105, A324201.

Cf. also A324057, A324187.

A106315(n) = (n*numdiv(n) % sigma(n));
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A156552(n) = if(1==n, 0, if(!(n%2), 1+(2*A156552(n/2)), 2*A156552(A064989(n))));
A324051(n) = A106315(A156552(n));

a(n) = A106315(A156552(n)).

a(n) = (A156552(n)*A324105(n)) mod A323243(n).

A057909 Numbers k such that 4^k + k is prime.

Original entry on oeis.org

1, 3, 9, 15, 37, 85, 133, 225, 1233, 12793, 108889

Offset: 1

Robert G. Wilson v, Nov 16 2000

a(11) > 20000. - Jinyuan Wang, Feb 01 2020

Cf. A052007, A057900.

Cf. A158879 (4^n + n).

Do[ If[ PrimeQ[ 4^n + n ], Print[ n ] ], {n, 0, 3000} ]

is(n)=ispseudoprime(4^n+n) \\ Charles R Greathouse IV, Jun 13 2017

a(10) from Jinyuan Wang, Feb 01 2020

a(11) from Hugo Pfoertner, Mar 02 2024

A129963 Primes of the form 4^k + k.

Original entry on oeis.org

5, 67, 262153, 1073741839, 18889465931478580854821, 1496577676626844588240573268701473812127674924007509, 118571099379011784113736688648896417641748464297615937576404566024103044751294597

Offset: 1

Cino Hilliard, Jun 10 2007, Aug 20 2007

nonn

It is convenient, although not necessary, to let k be an odd number since k even => 4^k + k is even > 2.
Conjecture: sequence is infinite.
The next term (a(8)) has 126 digits. - Harvey P. Dale, Jun 05 2014

			For k = 3, 4^3 + 3 = 67 is prime.

Amiram Eldar, Table of n, a(n) for n = 1..9

Cf. A057909, A158879.

Select[Table[4^n+n,{n,1,251,2}],PrimeQ] (* Harvey P. Dale, Jun 05 2014 *)

f(n) = for(x=1,n,y=2^x+x;if(isprime(y),print1(y",")))

a(n) = A158879(A057909(n)). - Amiram Eldar, Jul 04 2024

A175976 a(n) = 4^n - 3*n + 1.

Original entry on oeis.org

2, 2, 11, 56, 245, 1010, 4079, 16364, 65513, 262118, 1048547, 4194272, 16777181, 67108826, 268435415, 1073741780, 4294967249, 17179869134, 68719476683, 274877906888, 1099511627717, 4398046511042, 17592186044351, 70368744177596, 281474976710585, 1125899906842550

Offset: 0

Vincenzo Librandi, Nov 02 2010

			a(1)=4-3+1=2. a(2)=16-6+1=11.

Vincenzo Librandi, Table of n, a(n) for n = 0..250
Index entries for linear recurrences with constant coefficients, signature (6,-9,4).

Cf. A131098, A158879.

[4^n-3*n+1: n in [0..30]]; // Vincenzo Librandi, Mar 20 2014

A175976 := proc(n) 4^n-3*n+1 ; end proc:

Table[4^n-3n+1,{n,0,30}] (* or *) LinearRecurrence[{6,-9,4},{2,2,11},30] (* Harvey P. Dale, Jul 07 2013 *)

G.f.: (-2+10*x-17*x^2)/((4*x-1)*(x-1)^2).
From Bruno Berselli, Nov 04 2010: (Start)
a(n) - 6*a(n-1) + 9*a(n-2) - 4*a(n-3) = 0 for n > 2.
a(n) = A158879(n) - A131098(n+1) (n > 0). (End)
E.g.f.: exp(x)*(1 - 3*x + exp(3*x)). - Elmo R. Oliveira, Mar 07 2025

G.f., program and link to recurrences from R. J. Mathar, Nov 03 2010

cp's OEIS Frontend

A104745 a(n) = 5^n + n.

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A226199 a(n) = 7^n + n.

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A226201 a(n) = 8^n + n.

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A226202 a(n) = 9^n + n.

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A226200 a(n) = 6^n + n.

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Magma

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A226737 a(n) = 11^n + n.

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A324051 a(n) = A106315(A156552(n)).