A081552 -id:A081552 - 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

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

A081551 Triangle, read by rows, in which the n-th row contains n smallest n-digit numbers.

Original entry on oeis.org

1, 10, 11, 100, 101, 102, 1000, 1001, 1002, 1003, 10000, 10001, 10002, 10003, 10004, 100000, 100001, 100002, 100003, 100004, 100005, 1000000, 1000001, 1000002, 1000003, 1000004, 1000005, 1000006, 10000000, 10000001, 10000002, 10000003, 10000004, 10000005, 10000006, 10000007

Offset: 1

Amarnath Murthy, Apr 01 2003

This sequence has asymptotic density 0 and Banach density 1 (see Mithun Kumar Das reference p.2). - Franz Vrabec, Jul 28 2019

			Triangle begins as:
       1;
      10,     11;
     100,    101,    102;
    1000,   1001,   1002,   1003;
   10000,  10001,  10002,  10003,  10004;
  100000, 100001, 100002, 100003, 100004, 100005;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Mithun Kumar Das, Pramod Eyyunni, and Bhuwanesh Rao Patil, Sparse subsets of the natural numbers and Euler's totient function, arXiv:1907.09847v1 [math.NT] 23 Jul 2019.

Cf. A081552, A081553.

Table[10^(n-1) +k-1, {n,12}, {k,n}]//Flatten (* G. C. Greubel, May 27 2021 *)

flatten([[10^(n-1) +k-1 for k in (1..n)] for n in (1..12)]) # G. C. Greubel, May 27 2021

From Franz Vrabec, Jul 28 2019: (Start)
T(n, k) = 10^(n-1) + k - 1.
As a one-dimensional sequence: a(n) = 10^m + n - (m^2 + m + 2)/2 where m = floor((-1 + sqrt(8*n-7))/2). (End)

More terms from Philippe Deléham, Mar 28 2009

A081553 Sum of n-th row of A081551.

Original entry on oeis.org

1, 21, 303, 4006, 50010, 600015, 7000021, 80000028, 900000036, 10000000045, 110000000055, 1200000000066, 13000000000078, 140000000000091, 1500000000000105, 16000000000000120, 170000000000000136, 1800000000000000153, 19000000000000000171, 200000000000000000190, 2100000000000000000210, 22000000000000000000231

Offset: 1

Amarnath Murthy, Apr 01 2003

G. C. Greubel, Table of n, a(n) for n = 1..500
Index entries for linear recurrences with constant coefficients, signature (23,-163,361,-320,100).

Cf. A081551, A081552.

Cf. A126431, A161680.

Table[n*10^(n-1) +n(n-1)/2, {n,40}] (* G. C. Greubel, May 27 2021 *)

[n*10^(n-1) +binomial(n,2) for n in (1..40)] # G. C. Greubel, May 27 2021

a(n) = n*10^(n-1) + n*(n-1)/2.
From G. C. Greubel, May 27 2021: (Start)
G.f.: x*(1 -2*x -17*x^2 +99*x^3)/((1-x)^3 * (1-10*x)^2).
E.g.f.: (1/2)*x*( x*exp(x) + 2*exp(10*x) ). (End)

Terms a(13) onward added by G. C. Greubel, May 27 2021

A266959 Smallest n-digit number ending in n.

Original entry on oeis.org

1, 12, 103, 1004, 10005, 100006, 1000007, 10000008, 100000009, 1000000010, 10000000011, 100000000012, 1000000000013, 10000000000014, 100000000000015, 1000000000000016, 10000000000000017, 100000000000000018, 1000000000000000019, 10000000000000000020

Offset: 1

Wesley Ivan Hurt, Jan 09 2016

Digital sum of a(n) = digsum(n) + 1 for n>1.

3, 229, 4987 are the initial values of n for prime a(n). - Altug Alkan, Jan 17 2016

			a(4) = 1004 because it is the smallest 4-digit number ending in 4.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (12,-21,10).

Cf. A007953 (digsum), A081552, A279913.

[1] cat [n+10^(n-1): n in [2..30]]; // Vincenzo Librandi, Jan 10 2016

A266959:=n->n+10^(n-1): 1, seq(A266959(n), n=2..30);

Join[{1}, Table[n + 10^(n - 1), {n, 2, 20}]]

Vec(x*(1-20*x^2+10*x^3)/((1-x)^2*(1-10*x)) + O(x^30)) \\ Colin Barker, Jan 10 2016

a(n) = if(n==1, 1, n + 10^(n-1)); \\ Altug Alkan, Jan 17 2016

def A266959(n): return n+10**(n-1) if n > 1 else 1 # Chai Wah Wu, Jul 25 2022

a(n) = n + 10^(n-1) for n>1 with a(1) = 1.
a(n) = A081552(n) - 1 for n>1. - Michel Marcus, Jan 10 2016
From Colin Barker, Jan 10 2016: (Start)
a(n) = 12*a(n-1) - 21*a(n-2) + 10*a(n-3) for n>3.
G.f.: x*(1-20*x^2+10*x^3) / ((1-x)^2*(1-10*x)). (End)

cp's OEIS Frontend

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

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

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

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