A171553 -id:A171553 - cp's OEIS frontend

A100459 Values of n for which the decimal number 10...090...01 is an n-digit prime.

3, 11, 143, 623

Offset: 1

Harvey Dubner (harvey(AT)dubner.com), Nov 23 2004

a(5) > 84309. - Robert Price, Apr 14 2013

The number n must be of the form 2x+3, by definition. If x is odd, the number 10...090...01 is divisible by 11. Also, if x == 2 mod 6, it is divisible by 7. Thus, it must be either 4 mod 6 or 0 mod 6. Thus, 2*(6x+4) + 3 = 12x + 11 and 2*(6x+0) + 3 = 12x + 3, implying that a(n) must be congruent to 11 mod 12 or 3 mod 12. - Derek Orr, Jul 28 2014

			10000900001 is an 11-digit prime. Thus 11 is a member of this sequence. - _Derek Orr_, Jul 27 2014

Cf. A100028, A171553.

2Select[Range[1000], PrimeQ[100^# + 9 * 10^# + 1] &] + 1 (* Alonso del Arte, Jul 28 2014 *)

for(n=0,10^5,p=10^(2*n+2)+9*10^(n+1)+1;if(ispseudoprime(p),print1(2*n+3,", "))) \\ Derek Orr, Jul 27 2014

A225810 a(n) = (10^n)^2 + 4*(10^n) + 1.

Original entry on oeis.org

6, 141, 10401, 1004001, 100040001, 10000400001, 1000004000001, 100000040000001, 10000000400000001, 1000000004000000001, 100000000040000000001, 10000000000400000000001, 1000000000004000000000001, 100000000000040000000000001, 10000000000000400000000000001

Offset: 0

Lance J. Weingartz, Jul 29 2013

This is an instance of (10^n)^2 + x(10^n) + 1 which umbrellas A066138, A033934, A171375, A171410, A171461, A171513 and A171553 which all produce palindromes of the form 1...n...1 when n <> 0.

Colin Barker, Table of n, a(n) for n = 0..499
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. A033934, A066138, A171375, A171410, A171461, A171513, A171553.

Table[(10^n)^2 + 4*(10^n) + 1, {n, 0, 20}] (* T. D. Noe, Aug 12 2013 *)
LinearRecurrence[{111,-1110,1000},{6,141,10401},20] (* Harvey P. Dale, Oct 28 2017 *)

Vec(-3*(470*x^2-175*x+2)/((x-1)*(10*x-1)*(100*x-1)) + O(x^100)) \\ Colin Barker, Apr 27 2015

From Colin Barker, Apr 27 2015: (Start)
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3).
G.f.: -3*(470*x^2-175*x+2)/((x-1)*(10*x-1)*(100*x-1)). (End)
E.g.f.: exp(x)*(1 + 4*exp(9*x) + exp(99*x)). - Elmo R. Oliveira, Jul 04 2025

A225813 a(n) = (10^n)^2 + 7*(10^n) + 1.

Original entry on oeis.org

9, 171, 10701, 1007001, 100070001, 10000700001, 1000007000001, 100000070000001, 10000000700000001, 1000000007000000001, 100000000070000000001, 10000000000700000000001, 1000000000007000000000001, 100000000000070000000000001, 10000000000000700000000000001

Offset: 0

Lance J. Weingartz, Jul 29 2013

This is an instance of (10^n)^2 + x(10^n) + 1 which umbrellas A066138, A033934, A171375, A171410, A171461, A171513 and A171553 which all produce palindromes of the form 1...x...1 when n <> 0.

Colin Barker, Table of n, a(n) for n = 0..499
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. A033934, A066138, A171375, A171410, A171461, A171513, A171553.

A225813:=n->(10^n)^2 + 7*(10^n) + 1: seq(A225813(n), n=0..20); # Wesley Ivan Hurt, Apr 08 2017

Table[(10^n)^2 + 7*(10^n) + 1, {n, 0, 20}] (* T. D. Noe, Aug 12 2013 *)
LinearRecurrence[{111,-1110,1000},{9,171,10701},20] (* Harvey P. Dale, Apr 12 2020 *)

Vec(-9*(190*x^2-92*x+1)/((x-1)*(10*x-1)*(100*x-1)) + O(x^100)) \\ Colin Barker, Apr 27 2015

From Colin Barker, Apr 27 2015: (Start)
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3).
G.f.: -9*(190*x^2-92*x+1)/((x-1)*(10*x-1)*(100*x-1)). (End)
E.g.f.: exp(x)*(1 + 7*exp(9*x) + exp(99*x)). - Elmo R. Oliveira, Jul 04 2025

cp's OEIS Frontend

A100459 Values of n for which the decimal number 10...090...01 is an n-digit prime.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

A225810 a(n) = (10^n)^2 + 4*(10^n) + 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A225813 a(n) = (10^n)^2 + 7*(10^n) + 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula