A062397 a(n) = 10^n + 1.

2, 11, 101, 1001, 10001, 100001, 1000001, 10000001, 100000001, 1000000001, 10000000001, 100000000001, 1000000000001, 10000000000001, 100000000000001, 1000000000000001, 10000000000000001, 100000000000000001

Offset: 0

Author

Henry Bottomley, Jun 22 2001

Keywords

• easy
• nonn

Comments

The first three terms (indices 0, 1 and 2) are the only known primes. Moreover, the terms not of the form a(2^k) are all composite, except for a(0). Indeed, for all n >= 0, a(2n+1) is divisible by 11, a(4n+2) is divisible by 101, a(8n+4) is divisible by 73, a(16n+8) is divisible by 17, a(32n+16) is divisible by 353, a(64n+32) is divisible by 19841, etc. - M. F. Hasler, Nov 03 2018 [Edited based on the comment by Jeppe Stig Nielsen, Oct 17 2019]

This sequence also results when each term is generated by converting the previous term into a Roman numeral, then replacing each letter with its corresponding decimal value, provided that the vinculum is used and numerals are written in a specific way for integers greater than 3999, e.g., IV with a vinculum over the I and V for 4000. - Jamie Robert Creasey, Apr 14 2021

By Mihăilescu's theorem, a(n) can never be a perfect power (see "Catalan's conjecture" in Links). - Marco Ripà, Mar 10 2025

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Wikipedia, Catalan's conjecture.
• Index entries for linear recurrences with constant coefficients, signature (11,-10).

Crossrefs

Except for the initial term, essentially the same as A000533. Cf. A054977, A007395, A000051, A034472, A052539, A034474, A062394, A034491, A062395, A062396, A007689, A063376, A063481, A074600-A074624, A034524, A178248, A228081 for numbers one more than powers, i.e., this sequence translated from base n (> 2) to base 10.

Cf. A002283, A011557.

Cf. A038371 (smallest prime factor), A185121.

Programs

Magma

[10^n + 1: n in [0..35]]; // Vincenzo Librandi, Apr 30 2011

Mathematica

LinearRecurrence[{11, -10},{2, 11},18] (* Ray Chandler, Aug 26 2015 *)
10^Range[0,20]+1 (* Harvey P. Dale, Jan 21 2020 *)

PARI

a(n)=10^n+1 \\ Charles R Greathouse IV, Sep 24 2015

Formula

a(n) = 10*a(n-1) - 9 = A011557(n) + 1 = A002283(n) + 2.

From Mohammad K. Azarian, Jan 02 2009: (Start)

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

E.g.f.: exp(x) + exp(10*x). (End)