A054416 -id:A054416 - cp's OEIS frontend

A001562 Numbers n such that (10^n + 1)/11 is a prime.

5, 7, 19, 31, 53, 67, 293, 641, 2137, 3011, 268207, 1600787

Offset: 1

The a(10) to a(11) gap represents the largest relative gap seen so far in searching repunits with bases between -12 and 12. On average, there should have been 4 more primes added to this sequence by a(11), instead of just 1. - Paul Bourdelais, Feb 11 2010

J. Brillhart et al., Factorizations of b^n +- 1. Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 2nd edition, 1985; and later supplements.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

P. Bourdelais, A Generalized Repunit Conjecture, 2009.
J. Brillhart, Letter to N. J. A. Sloane, Aug 08 1970
J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
H. Dubner, Generalized repunit primes, Math. Comp., 61 (1993), 927-930. [Annotated scanned copy]
H. Dubner and T. Granlund, Primes of the Form (b^n+1)/(b+1), J. Integer Sequences, 3 (2000), #P00.2.7.
H. Lifchitz, Mersenne and Fermat primes field
S. S. Wagstaff, Jr., The Cunningham Project
Eric Weisstein's World of Mathematics, Repunit
R. G. Wilson, v, Letter to N. J. A. Sloane, circa 1991.

Equals 2*A054416 + 1.

Odd terms of A309358.

Select[Range[3000], PrimeQ[(10^# + 1) / 11] &] (* Vincenzo Librandi, Oct 29 2017 *)

isok(n) = (denominator(p=(10^n+1)/11)==1) && isprime(p); \\ Michel Marcus, Oct 29 2017

a(11) corresponds to a probable prime discovered by Paul Bourdelais, Feb 11 2010

a(12) corresponds to a probable prime discovered by Paul Bourdelais, May 04 2020

A095372 1+integers repeating "90" decimal digit pattern.

Original entry on oeis.org

1, 91, 9091, 909091, 90909091, 9090909091, 909090909091, 90909090909091, 9090909090909091, 909090909090909091, 90909090909090909091, 9090909090909090909091, 909090909090909090909091

Offset: 0

Labos Elemer, Jun 07 2004

These numbers arise for example as divisors of several repunits (A002275).
The aerated sequence A(n) = [1, 0, 91, 0, 9091, 0, 909091,...] is a divisibility sequence, i.e., A(n) divides A(m) whenever n divides m. It is the case P1 = 0, P2 = -11^2, Q = 10 of the 3-parameter family of 4th-order linear divisibility sequences found by Williams and Guy. - Peter Bala, Aug 22 2019
Except for a(0) = 1, these terms M are such that 21 * M = 1M1, where 1M1 denotes the concatenation of 1, M and 1. Actually 21 is A329914(1) and a(1) = A329915(1) = 91, and the terms >=91 form the set {M_21}; for example, 21 * 909091 = 1(909091)1. - Bernard Schott, Dec 01 2019

			Digit-pattern P=[ab..z] repeating integers equal formally with P*(-1+10^(Ln))/(-1+10^L), where L is the length of pattern;
a(9) divides A002275(38) repunit. See A095371.

H. C. Williams and R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory 7 (5) (2011) 1255-1277.
H. C. Williams and R. K. Guy, Some Monoapparitic Fourth Order Linear Divisibility Sequences Integers, Volume 12A (2012) The John Selfridge Memorial Volume.
Index entries for linear recurrences with constant coefficients, signature (101,-100).

Cf. A002275, A095371.
Cf. A001562, A015585, A054416, A097209, A100047, A152577.
Cf. A329914, A329915.

Mathematica
```
Table[1+90*(100^n-1)/99, {n, 0, 20}]
```

a(n) = 1 + 90*(-1 + 100^n)/99 = (10^(2*n+1) + 1)/11. - Rick L. Shepherd, Aug 01 2004
From Colin Barker, Jul 03 2013: (Start)
a(n) = 101*a(n-1) - 100*a(n-2).
G.f.: -(10*x-1)/((x-1)*(100*x-1)). (End)
E.g.f.: exp(x)*(1 + 10*(exp(99*x) - 1)/11). - Elmo R. Oliveira, Mar 15 2025

A097209 Primes of the form (10^p + 1)/11 (corresponding p are in A001562).

Original entry on oeis.org

9091, 909091, 909090909090909091, 909090909090909090909090909091, 9090909090909090909090909090909090909090909090909091, 909090909090909090909090909090909090909090909090909090909090909091

Offset: 1

Rick L. Shepherd, Jul 30 2004

nonn

Equivalently, primes of the form 9090...9091 (A054416(n)-1 copies of 90 followed by 91), the subsequence of all primes in A095372.

These primes appear in A187614 because the decimal representation of their reciprocal contains only the digits 0, 1, 8, and 9.

Cf. A001562, A054416, A095372.

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A001562 Numbers n such that (10^n + 1)/11 is a prime.

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A095372 1+integers repeating "90" decimal digit pattern.

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A097209 Primes of the form (10^p + 1)/11 (corresponding p are in A001562).

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