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

Author

Eric Angelini, Sep 13 2004

Keywords

• nonn
• nice
• base

Comments

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

Examples

			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
		

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

Crossrefs

Cf. A097487, A097968, A098099, A228595.

Programs

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 *)

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

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