A093166 -id:A093166

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A099422

Numbers k such that 8*R_k - 5 is prime, where R_k = 11...1 is the repunit (A002275) of length k.

+10
3

1, 2, 3, 5, 8, 9, 15, 51, 71, 77, 224, 296, 315, 2090, 2906, 3395, 3882, 5114, 6056, 7254, 7995, 18173, 18971, 35006, 69674, 175428, 253313

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Also numbers k >= 1 such that (8*10^k - 53)/9 is prime.

a(26) > 10^5. - Robert Price, Oct 31 2014

LINKS

Table of n, a(n) for n=1..27.

Makoto Kamada, Prime numbers of the form 88...883.

Index entries for primes involving repunits

FORMULA

a(n) = A056694(n) + 1.

MATHEMATICA

Do[ If[ PrimeQ[ 8(10^n - 1)/9 - 5], Print[n]], {n, 1, 5000}]

PROG

(Magma) [n: n in [1..500] | IsPrime((8*10^n-53) div 9)]; // Vincenzo Librandi, Nov 01 2014

CROSSREFS

Cf. A002275, A056664, A056694, A093166.

KEYWORD

nonn

AUTHOR

Robert G. Wilson v, Oct 14 2004

EXTENSIONS

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Jan 01 2008

a(22)-a(25) from Robert Price, Oct 31 2014

a(1)=0 removed by Georg Fischer, Jan 03 2021

a(26)-a(27) from Kamada data by Tyler Busby, Apr 16 2024

STATUS

approved

A056694

Numbers k such that 80*R_k + 3 is prime, where R_k = 11...1 is the repunit (A002275) of length k.

+10
2

0, 1, 2, 4, 7, 8, 14, 50, 70, 76, 223, 295, 314, 2089, 2905, 3394, 3881, 5113, 6055, 7253, 7994, 18172, 18970, 35005, 69673

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

Also numbers k such that (8*10^(k+1)-53)/9 is prime.

a(26) > 10^5. - Robert Price, Oct 31 2014

LINKS

Table of n, a(n) for n=1..25.

Makoto Kamada, Prime numbers of the form 88...883.

Index entries for primes involving repunits

FORMULA

a(n) = A099422(n) - 1. [adapted by Georg Fischer, Jan 04 2021]

MATHEMATICA

Do[ If[ PrimeQ[80*(10^n - 1)/9 + 3], Print[n]], {n, 0, 5000}]

PROG

(Magma) [n: n in [0..500] | IsPrime((8*10^(n+1)-53) div 9)]; // Vincenzo Librandi, Nov 01 2014

CROSSREFS

Cf. A002275, A093166, A099422.

KEYWORD

hard,nonn

AUTHOR

Robert G. Wilson v, Aug 10 2000

EXTENSIONS

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Jan 01 2008

a(22)-a(25) from Robert Price, Oct 31 2014

STATUS

approved

A173811

a(n) = (8*10^n - 53)/9 for n > 0.

+10
2

3, 83, 883, 8883, 88883, 888883, 8888883, 88888883, 888888883, 8888888883, 88888888883, 888888888883, 8888888888883, 88888888888883, 888888888888883, 8888888888888883, 88888888888888883, 888888888888888883, 8888888888888888883, 88888888888888888883, 888888888888888888883

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..100

Index entries for linear recurrences with constant coefficients, signature (11,-10).

FORMULA

a(n) = 10*a(n-1) + 53 for n > 0, a(0) = -5.

From Vincenzo Librandi, Jul 05 2012: (Start)

G.f.: x*(3+50*x)/((1-x)*(1-10*x)).

a(n) = 11*a(n-1) - 10*a(n-2). (End)

E.g.f.: exp(x)*(8*exp(9*x) - 53)/9. - Elmo R. Oliveira, Sep 09 2024

MATHEMATICA

NestList[10#+53&, 3, 20] (* Harvey P. Dale, Jun 13 2011 *)

CoefficientList[Series[(3+50*x)/((1-x)*(1-10*x)), {x, 0, 30}], x] (* Vincenzo Librandi, Jul 05 2012 *)

PROG

(Magma)[(8*10^n-53)/9: n in [1..20]]; // Vincenzo Librandi, Jul 05 2012

CROSSREFS

Cf. A093166.

KEYWORD

nonn,easy

AUTHOR

Vincenzo Librandi, Feb 25 2010

STATUS

approved

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