A098067 Consider the succession of single digits of the positive integers: 1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 1 4 1 5 1 6 ... (A007376). This sequence is the lexicographically earliest derangement of the positive integers that produces the same succession of digits.

12, 3, 4, 5, 6, 7, 8, 9, 10, 1, 112, 13, 14, 15, 16, 17, 18, 19, 20, 2, 122, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73

Offset

1,1

Comments

Derangement here means that a(n) != n.

Links

Paul Tek, Table of n, a(n) for n = 1..10000

Eric Angelini, Jeux de suites, in Dossier Pour La Science, pp. 32-35, Volume 59 (Jeux math'), April/June 2008, Paris.

Paul Tek, PERL program for this sequence

Example

We must begin with "1,2,3,..." and we cannot have a(1) = 1, so the first possible term is "12". The next term must be the smallest available positive integer not leading to a contradiction, thus "3"; the next one will be "4"; etc. After a(10) = 1, we cannot have a(11) = 11, so we use "112" instead. We are not allowed to use "2" after "19" because the next term would have a leading zero, which is forbidden. - Eric Angelini, Aug 12 2008

Mathematica

lim = 80; f[lst_List, k_] := Block[{L = lst, g, a = {}, m = 0}, g[] := {Set[m, First@ FromDigits@ Append[IntegerDigits@ m, First@ #]], Set[L, Last@ #]} &@ TakeDrop[L, 1]; Do[g[]; While[Or[m == Length@ a + 1, First@ L == 0, MemberQ[a, m]], g[]]; AppendTo[a, m]; m = 0, {k}]; a]; f[Flatten@ Map[IntegerDigits, Range@ lim], Floor[lim - 10^(Log10@ lim - 1)]] (* Michael De Vlieger, Nov 29 2015, Version 10.2 *)

Prog

(Perl) See Link section.
(Python)
from itertools import count
def diggen():
    for k in count(1): yield from list(map(int, str(k)))
def aupton(terms):
    g = diggen()
    alst, aset, _, _, nextd = [12], {12}, next(g), next(g), next(g)
    for n in range(2, terms+1):
        an, nextd = nextd, next(g)
        while an in aset or an == n or nextd == 0:
            an, nextd = int(str(an) + str(nextd)), next(g)
        alst.append(an); aset.add(an)
    return alst
print(aupton(72)) # Michael S. Branicky, Dec 03 2021

Crossrefs

Cf. A097487, A097968, A098099, A228595.

Sequence in context: A126860 A010203 A129197 * A070604 A268592 A127146

Adjacent sequences: A098064 A098065 A098066 * A098068 A098069 A098070

Keyword

nonn,nice,base

Author

Eric Angelini, Sep 13 2004

Extensions

Corrected and extended by Jacques ALARDET and Eric Angelini, Aug 12 2008

Derangement wording introduced by Danny Rorabaugh, Nov 26 2015

Edited by Danny Rorabaugh, Nov 29 2015

Status

approved