A104745 -id:A104745 - cp's OEIS frontend

A081552 Leading terms of rows in A081551.

1, 11, 102, 1003, 10004, 100005, 1000006, 10000007, 100000008, 1000000009, 10000000010, 100000000011, 1000000000012, 10000000000013, 100000000000014, 1000000000000015, 10000000000000016, 100000000000000017, 1000000000000000018

Offset: 1

Amarnath Murthy, Apr 01 2003

More generally, a(n) = B^K + n; K = floor(log_B a(n-1)) + 1. This sequence has B=10, a(0)=1; A006127 has B=2, a(0)=1; A052944 has B=2, a(0)=2; A104743 has B=3, a(0)=1; A104745 has B=5, a(0)=1. - Ctibor O. Zizka, Mar 22 2008

Vincenzo Librandi, Table of n, a(n) for n = 1..300
Index entries for linear recurrences with constant coefficients, signature (12,-21,10).

Cf. A011557, A081551, A081553, A085952 (first differences, after n=2).

[10^(n-1)+n-1: n in [1..20]]; // Vincenzo Librandi, Jun 16 2013

I:=[1, 11, 102]; [n le 3 select I[n] else 12*Self(n-1)-21*Self(n-2)+10*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Jun 16 2013

seq(10^(n-1) +n-1, n=1..40); # G. C. Greubel, May 27 2021

Table[10^(n-1) +n-1, {n,30}] (* or *) CoefficientList[Series[(1-x-9x^2)/((1-10x)(1-x)^2), {x, 0, 30}], x]  (* Vincenzo Librandi, Jun 16 2013 *)

[10^(n-1) +n-1 for n in (1..40)] # G. C. Greubel, May 27 2021

a(n) = 10^(n-1) + n-1.
G.f.: x*(1 -x -9*x^2)/((1-10*x)*(1-x)^2). - Vincenzo Librandi, Jun 16 2013
a(n) = 12*a(n-1) -21*a(n-2) +10*a(n-3). - Vincenzo Librandi, Jun 16 2013
E.g.f.: (1/10)*(9 - 10*(1-x)*exp(x) + exp(10*x)). - G. C. Greubel, May 27 2021

A158879 a(n) = 4^n + n.

Original entry on oeis.org

1, 5, 18, 67, 260, 1029, 4102, 16391, 65544, 262153, 1048586, 4194315, 16777228, 67108877, 268435470, 1073741839, 4294967312, 17179869201, 68719476754, 274877906963, 1099511627796, 4398046511125, 17592186044438, 70368744177687

Offset: 0

Philippe Deléham, Mar 28 2009

			a(0)=4^0+0 = 1, a(1)=4^1+1 = 5, a(2)=4^2+2 = 18, a(3)=4^3+3 = 67, ...

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Index entries for linear recurrences with constant coefficients, signature (6,-9,4).

Cf. A006127, A081552, A104743, A104745.

List([0..30], n-> n+4^n); # G. C. Greubel, Mar 04 2020

[4^n+n: n in [0..30]]; // Vincenzo Librandi, Jun 16 2013

seq( 4^n+n, n=0..30); # G. C. Greubel, Mar 04 2020

Table[4^n+n,{n,0,40}] (* Vladimir Joseph Stephan Orlovsky, May 19 2011 *)
CoefficientList[Series[(1-x-3x^2)/((1-4x)(1-x)^2), {x, 0, 30}], x] (* Vincenzo Librandi, Jun 16 2013 *)
LinearRecurrence[{6,-9,4},{1,5,18},30] (* Harvey P. Dale, Jun 02 2016 *)

a(n)=4^n+n \\ Charles R Greathouse IV, Oct 07 2015

[n+4^n for n in (0..30)] # G. C. Greubel, Mar 04 2020

G.f.: (1 - x - 3*x^2)/((1-4*x)*(1-x)^2). - R. J. Mathar, Mar 29 2009
a(n) = 6*a(n-1) -9*a(n-2) +4*a(n-3). - R. J. Mathar, Mar 29 2009
E.g.f.: x*exp(x) + exp(4*x). - G. C. Greubel, Mar 04 2020

Corrected typo in a(22) from R. J. Mathar, Mar 29 2009

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

A362555 Number of distinct n-digit suffixes generated by iteratively multiplying an integer by 6, where the initial integer is 1.

Original entry on oeis.org

2, 7, 28, 129, 630, 3131, 15632, 78133, 390634, 1953135, 9765636, 48828137, 244140638, 1220703139, 6103515640, 30517578141, 152587890642, 762939453143, 3814697265644, 19073486328145, 95367431640646, 476837158203147, 2384185791015648, 11920928955078149, 59604644775390650

Offset: 1

Gil Moses, Apr 24 2023

			For n = 2, we begin with 1, iteratively multiply by 6 and count the terms before the last 2 digits begin to repeat. We obtain 1, 6, 36, 216, 1296, 7776, 46656, ... . The next term is 279936, which repeats the last 2 digits 36. Thus, the number of distinct terms is a(2) = 7.

Paolo Xausa, Table of n, a(n) for n = 1..1000
Wikipedia, Multiplicative Order
Index entries for linear recurrences with constant coefficients, signature (7,-11,5).

Cf. A104745, A370557.

Cf. A362468 (with 4 as the multiplier).

A362555[n_]:=5^(n-1)+n;Array[A362555,30] (* Paolo Xausa, Nov 18 2023 *)

a(n) = 5^(n-1) + n.
From Stefano Spezia, Apr 27 2023: (Start)
O.g.f.: (1 - 5*x + 4*x^2 - 4*x^3)/((1 - x)^2*(1 - 5*x)).
E.g.f.: (4 + exp(5*x) + 5*exp(x)*x)/4. (End)

A181572 Number of distinct prime divisors of 5^n + n.

Original entry on oeis.org

2, 1, 1, 2, 3, 3, 4, 2, 3, 4, 3, 3, 3, 5, 5, 3, 2, 5, 4, 4, 4, 3, 2, 3, 5, 3, 4, 4, 5, 5, 4, 3, 3, 4, 4, 3, 5, 5, 4, 9, 3, 6, 6, 5, 4, 6, 3, 4, 5, 4, 3, 5, 5, 4, 9, 2, 5, 4, 4, 6, 7, 4, 3, 6, 7, 4, 7, 4, 6, 5, 4, 4, 7, 7, 4, 4, 5, 6, 8, 6, 3, 3, 6, 3, 9, 5, 4

Offset: 1

Michel Lagneau, Jan 30 2011

nonn

5^n + n is composite for all n < 1000. - Pete L. Clark ( Department of Mathematics University of Georgia).

5^7954 + 7954 has been found to be prime by a computer (see Links).

			a(5) = 3 because 5^5 + 5 = 2*5*313 has 3 prime factors.

Amiram Eldar, Table of n, a(n) for n = 1..182
5^n+n is never prime? 5^n+n is never prime?
Mersenneforum.org discussion, 5^7954 + 7954 has been found to be prime by a computer

Cf. A001221, A104745.

[#PrimeDivisors(5^n+n): n in [1..107]]; // Marius A. Burtea, Feb 19 2020

with(numtheory):for n from 1 to 100 do:x:=5^n + n: y:=nops(factorset(x)):printf(`%d,  `,y):od:

PrimeNu/@Table[5^n+n,{n,20}] (* Harvey P. Dale, Feb 17 2013 *)

a(n)=omega(5^n+n) \\ Charles R Greathouse IV, Aug 27 2014

a(n) = A001221(A104745(n)). - Amiram Eldar, Feb 19 2020

a(71)-a(87) from Amiram Eldar, Feb 19 2020

A370557 Numbers k such that A362555(k) = (5^k + 5*k)/5 is prime.

Original entry on oeis.org

1, 2, 1014, 16058

Offset: 1

Hugo Pfoertner, Feb 24 2024

If it exists, a(5) > 50000.

Cf. A104745, A362555, A370190.

PARI
```
is(n) = ispseudoprime(5^(n-1)+n)
```

cp's OEIS Frontend

A081552 Leading terms of rows in A081551.

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A158879 a(n) = 4^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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Magma

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

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Magma

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

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