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%I A001292 #79 Jul 01 2022 22:04:15
%S A001292 1,12,21,123,231,312,1234,2341,3412,4123,12345,23451,34512,45123,
%T A001292 51234,123456,234561,345612,456123,561234,612345,1234567,2345671,
%U A001292 3456712,4567123,5671234,6712345,7123456
%N A001292 Concatenations of cyclic permutations of initial positive integers.
%C A001292 Entries are sorted numerically, so after a(45) = 912345678 we have a(46) = 10123456789 instead of a(46) = 12345678910. - _Giovanni Resta_, Mar 21 2017
%C A001292 From _Marco Ripà_, Apr 21 2022: (Start)
%C A001292 In 1996, Kenichiro Kashihara conjectured that there is no prime power of an integer (A093771) belonging to this sequence (disregarding the trivial case 1); a direct search from 12 to a(100128) has confirmed the conjecture up to 10^1035. There are no perfect powers among terms t which are permutations of 123_...(m - 1)_m for m == {2, 3, 5, 6} (mod 9). This is since 10 == 1 (mod 9) and also (1 + 0) == 1 (mod 9), so digit position has no effect. Hence, t == A134804(m) (mod 9). Now, if m is such that A134804(m) = {3, 6}, there is a lone factor of 3, which is not a perfect power (indeed).
%C A001292 Therefore, any perfect power in this sequence is necessarily congruent modulo 9 to 0 or 1.
%C A001292 (End)
%H A001292 John Cerkan, <a href="/A001292/b001292.txt">Table of n, a(n) for n = 1..10000</a>
%H A001292 Marco Ripà, <a href="https://doi.org/10.13140/RG.2.2.30313.77921/3">On some open problems concerning perfect powers</a>, ResearchGate (2022).
%H A001292 Florentin Smarandache, <a href="http://www.gallup.unm.edu/~smarandache/OPNS.pdf">Only Problems, Not Solutions!</a>, Unsolved Problem #16, p. 18.
%t A001292 Sort@ Flatten@ Table[ FromDigits[ Join @@ IntegerDigits /@ RotateLeft[Range[n], i - 1]], {n, 11}, {i, n}] (* _Giovanni Resta_, Mar 21 2017 *)
%o A001292 (Python)
%o A001292 from itertools import count, islice
%o A001292 def A001292gen():
%o A001292     s = []
%o A001292     for i in count(1):
%o A001292         s.append(str(i))
%o A001292         yield from sorted(int("".join(s[j:]+s[:j])) for j in range(i))
%o A001292 print(list(islice(A001292gen(), 46))) # _Michael S. Branicky_, Jul 01 2022
%Y A001292 Cf. A093771, A134804, A352991.
%K A001292 nonn,base
%O A001292 1,2
%A A001292 R. Muller

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