cp's OEIS Frontend

A319023 Let S be the sequence generated by these rules: 1 is in S, and if k is in S, then A057168(k) and A319021(k) are in S, and duplicates are deleted as they occur; a(n) is the n-th term of S.

%I A319023 #13 Sep 09 2018 12:41:44
%S A319023 1,2,3,4,5,9,8,6,7,10,27,16,14,11,12,29,81,32,20,19,13,17,18,30,31,82,
%T A319023 243,64,34,24,22,21,15,23,28,39,36,47,33,84,245,729,128,66,38,25,35,
%U A319023 43,45,40,54,55,49,37,88,90,246,247,730,2187,256,130,68,72,41
%N A319023 Let S be the sequence generated by these rules: 1 is in S, and if k is in S, then A057168(k) and A319021(k) are in S, and duplicates are deleted as they occur; a(n) is the n-th term of S.
%C A319023 This sequence is a permutation of the natural numbers (with inverse A319024):
%C A319023 - this sequence is injective,
%C A319023 - this sequence is surjective: by contradiction:
%C A319023 - let m be the least integer missing from the sequence,
%C A319023 - as a(1) = 1, we have m > 1,
%C A319023 - also, m belongs to A000225 and to A062318,
%C A319023 - however the only positive integer belonging to both sequences is 1,
%C A319023 - hence a contradiction, QED.
%H A319023 Rémy Sigrist, <a href="/A319023/b319023.txt">Table of n, a(n) for n = 1..10000</a>
%H A319023 Rémy Sigrist, <a href="/A319023/a319023.gp.txt">PARI program for A319023</a>
%H A319023 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%e A319023 The first terms, alongside b(n) = A057168(a(n)) and t(n) = A319021(a(n)), are:
%e A319023   n   a(n)  b(n)  t(n)
%e A319023   --  ----  ----  ----
%e A319023    1     1     2     3
%e A319023    2     2     4     4
%e A319023    3     3     5     9
%e A319023    4     4     8     6
%e A319023    5     5     6     7
%e A319023    6     9    10    27
%e A319023    7     8    16    14
%e A319023    8     6     9    10
%e A319023    9     7    11    11
%e A319023   10    10    12    12
%e A319023   11    27    29    81
%e A319023   12    16    32    20
%e A319023   13    14    19    16
%e A319023   14    11    13    13
%e A319023   15    12    17    18
%e A319023   16    29    30    31
%e A319023   17    81    82   243
%e A319023   18    32    64    34
%e A319023   19    20    24    22
%e A319023   20    19    21    21
%o A319023 (PARI) See Links section.
%Y A319023 Cf. A000225, A057168, A062318, A319021, A319024 (inverse).
%K A319023 nonn,base,look
%O A319023 1,2
%A A319023 _Rémy Sigrist_, Sep 08 2018