A226202 -id:A226202 - cp's OEIS frontend

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

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

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

A221911 a(n) = 9^n + 9*n.

Original entry on oeis.org

1, 18, 99, 756, 6597, 59094, 531495, 4783032, 43046793, 387420570, 3486784491, 31381059708, 282429536589, 2541865828446, 22876792455087, 205891132094784, 1853020188851985, 16677181699666722, 150094635296999283, 1350851717672992260, 12157665459056928981

Offset: 0

Vincenzo Librandi, Mar 03 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (11,-19,9).

Cf. A198399, A226202, A370663.

Magma
```
[9^n + 9*n: n in [0..25]];
```

I:=[1, 18, 99]; [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 + 9 n), {n, 0, 25}] (* or *) CoefficientList[Series[(1 + 7 x - 80 x^2)/((1 - x)^2 (1 - 9 x)), {x, 0, 30}], x]
LinearRecurrence[{11,-19,9},{1,18,99},20] (* Harvey P. Dale, Aug 18 2014 *)

a(n)=9^n+9*n \\ Charles R Greathouse IV, Apr 18 2013

G.f.: (1+7*x-80*x^2)/((1-x)^2*(1-9*x)).
a(n) = 11*a(n-1) - 19*a(n-2) + 9*a(n-3).
E.g.f.: exp(x)*(exp(8*x) + 9*x). - Elmo R. Oliveira, Sep 09 2024

A370663 Numbers k such that (9^k + 9*k)/9 is prime.

Original entry on oeis.org

1, 2, 4, 8, 10, 28, 1348, 1424, 12838

Offset: 1

Hugo Pfoertner, Feb 27 2024

If it exists, a(10) > 50000.

Cf. A221911, A226202.

[n: n in [1..130] |IsPrime ((9^n + 9*n) div 9)]; // Vincenzo Librandi, Jul 04 2025

Select[Range[1,13000],PrimeQ[((9^#+9*#) / 9)]&] (* Vincenzo Librandi, Jul 04 2025 *)

PARI
```
is(n) = ispseudoprime(9^(n-1)+n)
```

cp's OEIS Frontend

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

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

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Magma

Magma

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

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Magma

Magma

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

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Magma

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A221911 a(n) = 9^n + 9*n.

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Magma

Magma

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A370663 Numbers k such that (9^k + 9*k)/9 is prime.

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PARI