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

0,1

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

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

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

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)

Mathematica

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

Prog

(Magma) [10^n + 1: n in [0..35]]; // Vincenzo Librandi, Apr 30 2011
(PARI) a(n)=10^n+1 \\ Charles R Greathouse IV, Sep 24 2015

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.

Sequence in context: A368879 A254320 A115062 * A158578 A003617 A114018

Adjacent sequences: A062394 A062395 A062396 * A062398 A062399 A062400

Keyword

easy,nonn

Author

Henry Bottomley, Jun 22 2001

Status

approved