cp's OEIS Frontend

A352991 Concatenation of all the distinct permutations of the first 1, 2, 3, ... (strictly) positive integers, arranged in ascending numerical order.

%I A352991 #41 Apr 17 2022 21:35:05
%S A352991 1,12,21,123,132,213,231,312,321,1234,1243,1324,1342,1423,1432,2134,
%T A352991 2143,2314,2341,2413,2431,3124,3142,3214,3241,3412,3421,4123,4132,
%U A352991 4213,4231,4312,4321,12345,12354,12435,12453,12534,12543,13245,13254,13425
%N A352991 Concatenation of all the distinct permutations of the first 1, 2, 3, ... (strictly) positive integers, arranged in ascending numerical order.
%C A352991 This sequence differs from A030299 starting from a(409114) = 10123456789. All the permutations are listed once and only once (e.g., the concatenation of the permutations of the elements of the set {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11} originates the number 1112345678910 which is a unique element of this sequence and appears only once, since 1_11_23456789 = 11_1_23456789 = 1112345678910).
%C A352991 A001292 is a subsequence of the present sequence. An open problem, published by Kenichiro Kashihara in 1996 (see References, p. 25, #30, Problem 2), is to find how many terms of A001292 (which is a subsequence of A030299) are powers of integers; Kashihara conjectured that there are none (even if, clearly, A001292(1) = 1 should be disregarded in order to keep the conjecture alive). Currently, only the terms up to the prime a(409120) = 10123457689 have been directly checked by the author of this sequence and no nontrivial perfect power has been found. On the other hand, many (maybe infinitely many) terms of the present sequence are nontrivial powers of integers (e.g., A352329(2) to A352329(36) are squares of integers and belong to this sequence).
%C A352991 Although A181129 is a subsequence of the present one, so that A181129(1) = a(19) = 2341, a(14) is the smallest prime in this sequence.
%C A352991 The number of digits of a(n) comes from A058183. There are exactly k! (Cf. A000142) terms having A058183(k) digits. - _David A. Corneth_, Apr 17 2022
%D A352991 Kenichiro Kashihara, Comments and Topics on Smarandache Notions and Problems, 25. Erhus University Press, Arizona, 1996. ISBN: 1-879585-55-3.
%H A352991 <a href="/index/Se#sequences_which_agree_for_a_long_time">Index entries for sequences which agree for a long time but are different</a>
%e A352991 a(3) = 21, since the number of permutations of {1, 2} is 2! = 2 and the concatenation 1_2 is smaller than 2_1 (while {1} originates only a(1) = 1, so that a(2) = 21).
%o A352991 (Python)
%o A352991 from itertools import count, islice, permutations
%o A352991 def agen(): # generator of terms
%o A352991     for k in count(1):
%o A352991         s = (int("".join(map(str, p))) for p in permutations(range(1, k+1)))
%o A352991         yield from sorted(set(s))
%o A352991 print(list(islice(agen(), 42))) # _Michael S. Branicky_, Apr 16 2022
%Y A352991 Cf. A000142, A001292, A058183, A030299, A181129.
%K A352991 nonn,easy,base
%O A352991 1,2
%A A352991 _Marco Ripà_, Apr 16 2022