A126431 -id:A126431 - cp's OEIS frontend

A064748 a(n) = n*10^n + 1.

1, 11, 201, 3001, 40001, 500001, 6000001, 70000001, 800000001, 9000000001, 100000000001, 1100000000001, 12000000000001, 130000000000001, 1400000000000001, 15000000000000001, 160000000000000001, 1700000000000000001, 18000000000000000001, 190000000000000000001

Offset: 0

N. J. A. Sloane, Oct 19 2001

Number of digits in (10^n)^(10^n) in base 10. - Altug Alkan, Apr 25 2016

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Paul Leyland, Cullen and Woodall numbers and their generalization to other bases.
Index entries for linear recurrences with constant coefficients, signature (21,-120,100).

Cf. A002064, A126431.

[ n*10^n+1: n in [0..20]]; // Vincenzo Librandi, Sep 16 2011

Table[n 10^n + 1, {n, 0, 18}] (* Michael De Vlieger, Apr 25 2016 *)

a(n) = n*10^n + 1; \\ Altug Alkan, Apr 25 2016

From Ilya Gutkovskiy, Apr 25 2016: (Start)
O.g.f.: (1 - 10*x + 90*x^2)/((1 - x)*(1 - 10*x)^2).
E.g.f.: (1 + 10*x*exp(9*x))*exp(x). (End)
From Elmo R. Oliveira, May 03 2025: (Start)
a(n) = 21*a(n-1) - 120*a(n-2) + 100*a(n-3).
a(n) = A126431(n) + 1. (End)

A064756 a(n) = n*10^n - 1.

Original entry on oeis.org

9, 199, 2999, 39999, 499999, 5999999, 69999999, 799999999, 8999999999, 99999999999, 1099999999999, 11999999999999, 129999999999999, 1399999999999999, 14999999999999999, 159999999999999999, 1699999999999999999, 17999999999999999999, 189999999999999999999, 1999999999999999999999

Offset: 1

N. J. A. Sloane, Oct 19 2001

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Paul Leyland, Factors of Cullen and Woodall numbers.
Paul Leyland, Generalized Cullen and Woodall numbers.
Amelia Carolina Sparavigna, Some Groupoids and their Representations by Means of Integer Sequences, International Journal of Sciences (2019) Vol. 8, No. 10.
Index entries for linear recurrences with constant coefficients, signature (21,-120,100).

Cf. for a(n) = n*k^n - 1: -A000012 (k=0), A001477 (k=1), A003261 (k=2), A060352 (k=3), A060416 (k=4), A064751 (k=5), A064752 (k=6), A064753 (k=7), A064754 (k=8), A064755 (k=9), this sequence (k=10), A064757 (k=11), A064758 (k=12).

Cf. A064748, A126431.

[ n*10^n-1: n in [1..20]]; // Vincenzo Librandi, Sep 16 2011

k:= 10; f:= gfun:-rectoproc({1 + (k-1)*n + k*n*a(n-1) - (n-1)*a(n) = 0, a(1) = k-1}, a(n), remember): map(f, [$1..20]); # Georg Fischer, Feb 19 2021

Array[# 10^# - 1 &, 18] (* Michael De Vlieger, Jan 14 2020 *)

From Elmo R. Oliveira, Sep 07 2024: (Start)
G.f.: x*(100*x^2 - 10*x - 9)/((x - 1)*(10*x - 1)^2).
E.g.f.: 1 + exp(x)*(10*x*exp(9*x) - 1).
a(n) = 21*a(n-1) - 120*a(n-2) + 100*a(n-3) for n > 3.
a(n) = A126431(n) - 1 = A064748(n) - 2. (End)

A374641 Decimal expansion of log(9/10), negated.

Original entry on oeis.org

1, 0, 5, 3, 6, 0, 5, 1, 5, 6, 5, 7, 8, 2, 6, 3, 0, 1, 2, 2, 7, 5, 0, 0, 9, 8, 0, 8, 3, 9, 3, 1, 2, 7, 9, 8, 3, 0, 6, 1, 2, 0, 3, 7, 2, 9, 8, 3, 2, 7, 4, 0, 7, 2, 5, 6, 3, 9, 3, 9, 2, 3, 3, 6, 9, 2, 5, 8, 4, 0, 2, 3, 2, 4, 0, 1, 3, 4, 5, 4, 6, 4, 8, 8, 7, 6, 5, 6, 9, 5

Offset: 0

Paolo Xausa, Jul 15 2024

Bailey et al. (1997) use Li_1(1/10) (see Formula section) to compute the ten billionth digit of this constant.
Bailey and Crandall (2001), p. 185, present this constant as an example of an irrational number that, provided their "Hypothesis A" (p. 176) is true, is normal to base 10.
Also decimal expansion of log(10/9). - Charles R Greathouse IV, Jul 17 2024

			0.105360515657826301227500980839312798306120372983...

Paolo Xausa, Table of n, a(n) for n = 0..10000
David Bailey, Peter Borwein, and Simon Plouffe, On the Rapid Computation of Various Polylogarithmic Constants, Mathematics of Computation, Vol. 66, No. 218, April 1997, pp. 903-913.
David H. Bailey and Richard E. Crandall, On the Random Character of Fundamental Constant Expansions, Experimental Mathematics, Vol. 10 (2001), Issue 2, pp. 175-190 (preprint draft).
Eric Weisstein's MathWorld, Polylogarithm.
Wikipedia, Polylogarithm.
Index entries for transcendental numbers

Cf. A126431, A374642, A374643, A374644.

Mathematica
```
First[RealDigits[Log[9/10], 10, 100]]
```

-log(.9) \\ Charles R Greathouse IV, Jul 17 2024

Equals Li_1(1/10) = Sum_{k >= 1} 1/(k*10^k), where Li_m(z) is the polylogarithm function. See Bailey et al. (1997), p. 909 and Bailey and Crandall (2001), p. 185.

Equals Integral_{x=0..1} (x^(1/3) - x^(1/5))/log(x) dx. - Kritsada Moomuang, May 27 2025

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

A126435 Primes of the form n^7-n-1.

Original entry on oeis.org

2097143, 1801088519, 21869999969, 42618442943, 78364164059, 137231006639, 194754273839, 435817657169, 678223072799, 1174711139783, 1727094849479, 3938980639103, 4398046511039, 4902227890559, 6722988818363, 19203908986079

Offset: 1

Artur Jasinski, Dec 26 2006

nonn

All terms end in 3 or 9. - Robert Israel, Jul 22 2019

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A002327, A002328, A116581, A126420, A126421, A126422, A126423, A126424, A126425, A126426, A126427, A126427, A126428, A126429, A126430, A126431, A126432 A126433, A126434.

map(t -> t^7-t-1, select(t -> isprime(t^7-t-1), [$1..10^4])); # Robert Israel, Jul 22 2019

k = 7; a = {}; Do[If[PrimeQ[x^k - x - 1], AppendTo[a, x^k - x - 1]], {x, 1, 100}]; a
Select[Table[n^7-n-1,{n,80}],PrimeQ] (* Harvey P. Dale, Jun 20 2020 *)

A126437 Primes of the form k^8-k-1.

Original entry on oeis.org

1679609, 5764793, 99999989, 4294967279, 282429536453, 377801998307, 5352009260441, 16815125390579, 39062499999949, 72301961339081, 83733937890569, 281474976710591, 513798374428571, 1113034787454899

Offset: 1

Artur Jasinski, Dec 26 2006

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A002327, A002328, A116581, A126420, A126421, A126422, A126423, A126424, A126425, A126426, A126427, A126427, A126428, A126429, A126430, A126431, A126432 A126433, A126434, A126434.

k = 8; a = {}; Do[If[PrimeQ[x^k - x - 1], AppendTo[a, x^k - x - 1]], {x, 1, 100}]; a
Select[Table[k^8-k-1,{k,80}],PrimeQ] (* Harvey P. Dale, Nov 06 2021 *)

A158749 a(n) = n*9^n.

Original entry on oeis.org

0, 9, 162, 2187, 26244, 295245, 3188646, 33480783, 344373768, 3486784401, 34867844010, 345191655699, 3389154437772, 33044255768277, 320275094369454, 3088366981419735, 29648323021629456, 283512088894331673, 2701703435345984178, 25666182635786849691, 243153309181138576020

Offset: 0

Zerinvary Lajos, Mar 25 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (18,-81).

Cf. A001019, A018215, A036289, A036290, A036291, A036292, A036293, A036294.

Cf. A038299, A126431.

[n*9^n: n in [0..20]]; // Vincenzo Librandi, Feb 25 2014

Table[n 9^n, {n, 0, 20}] (* Vincenzo Librandi, Feb 25 2014 *)

a(n) = n*9^n; \\ Joerg Arndt, Feb 23 2014

a(n) = n*9^n.
From R. J. Mathar, Mar 26 2009: (Start)
a(n) = 18*a(n-1) - 81*a(n-2) = A038299(n,1).
G.f.: 9*x/(1-9*x)^2. (End)
a(n) = A001019(n)*n. - Omar E. Pol, Mar 26 2009
From Amiram Eldar, Jul 20 2020: (Start)
Sum_{n>=1} 1/a(n) = log(9/8).
Sum_{n>=1} (-1)^(n+1)/a(n) = log(10/9). (End)
E.g.f.: 9*x*exp(9*x). - Elmo R. Oliveira, Sep 09 2024

A126438 Primes of the form n^9-n-1.

Original entry on oeis.org

509, 262139, 10077689, 387420479, 68719476719, 118587876479, 1207269217769, 7625597484959, 10578455953379, 129961739795039, 327381934393919, 1628413597910399, 1953124999999949, 5416169448144839, 10077695999999939

Offset: 1

Artur Jasinski, Dec 26 2006

nonn

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A002327, A002328, A116581, A126420, A126421, A126422, A126423, A126424, A126425, A126426, A126427, A126427, A126428, A126429, A126430, A126431, A126432 A126433, A126434, A126435, A126436, A126437.

k = 9; a = {}; Do[If[PrimeQ[x^k - x - 1], AppendTo[a, x^k - x - 1]], {x, 1, 100}]; a
Select[Table[n^9-n-1,{n,100}],PrimeQ] (* Harvey P. Dale, Mar 09 2016 *)

A126439 Least prime of the form x^n-x-1.

Original entry on oeis.org

5, 5, 13, 29, 61, 2097143, 1679609, 509, 1021, 8589934583, 4093, 67108859, 16381, 470184984569, 4294967291, 2218611106740436979, 68719476731, 1350851717672992079, 1048573, 10460353199, 4194301, 20013311644049280264138724244295359, 16777213, 108347059433883722041830239, 20282409603651670423947251285999, 58149737003040059690390159, 72057594037927931, 536870909, 999999999999999999999999999989

Offset: 2

Artur Jasinski, Dec 26 2006, Jan 19 2007

nonn

Cf. A002327, A002328, A116581, A126420, A126421, A126422, A126423, A126424, A126425, A126426, A126427, A126427, A126428, A126429, A126430, A126431, A126432 A126433, A126434, A126435, A126436, A126437, A126438.

a = {}; Do[k = 2; While[ ! PrimeQ[k^n -k - 1], k++ ]; AppendTo[a, k^n - k - 1], {n, 2, 30}]; a (*Artur Jasinski*)

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A064748 a(n) = n*10^n + 1.

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A064756 a(n) = n*10^n - 1.

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A374641 Decimal expansion of log(9/10), negated.

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A081553 Sum of n-th row of A081551.

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A126435 Primes of the form n^7-n-1.

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Maple

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A126437 Primes of the form k^8-k-1.

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A158749 a(n) = n*9^n.

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A126438 Primes of the form n^9-n-1.

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