A181129 -id:A181129 - 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.

1, 12, 21, 123, 132, 213, 231, 312, 321, 1234, 1243, 1324, 1342, 1423, 1432, 2134, 2143, 2314, 2341, 2413, 2431, 3124, 3142, 3214, 3241, 3412, 3421, 4123, 4132, 4213, 4231, 4312, 4321, 12345, 12354, 12435, 12453, 12534, 12543, 13245, 13254, 13425

Offset: 1

Marco Ripà, Apr 16 2022

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).
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).
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.
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

			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).

Kenichiro Kashihara, Comments and Topics on Smarandache Notions and Problems, 25. Erhus University Press, Arizona, 1996. ISBN: 1-879585-55-3.

Index entries for sequences which agree for a long time but are different

Cf. A000142, A001292, A058183, A030299, A181129.

from itertools import count, islice, permutations
def agen(): # generator of terms
    for k in count(1):
        s = (int("".join(map(str, p))) for p in permutations(range(1, k+1)))
        yield from sorted(set(s))
print(list(islice(agen(), 42))) # Michael S. Branicky, Apr 16 2022

A353025 Terms of A352991 which are perfect powers.

Original entry on oeis.org

1, 13527684, 34857216, 65318724, 73256481, 81432576, 139854276, 152843769, 157326849, 215384976, 245893761, 254817369, 326597184, 361874529, 375468129, 382945761, 385297641, 412739856, 523814769, 529874361, 537219684, 549386721, 587432169, 589324176, 597362481, 615387249, 627953481, 653927184

Offset: 1

Marco Ripà, Apr 17 2022

It appears that all terms are terms of A062503.
We note that a(n)=A352329(n) up to a(36)=A352329(36)=923187456, while the mentioned match does not hold starting from a(37)=14102987536 (since A352329(37)=1234608769).
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).
Therefore, all terms are necessarily congruent modulo 9 to 0 or 1 (see Marco Ripà link).
All terms up to 10^34 are squares (in particular, there are 67 squares with no more than 17 digits). - Aldo Roberto Pessolano, May 12 2022

			75910168324 is a term since 75910168324 = 275518^2.

Aldo Roberto Pessolano, Table of n, a(n) for n = 1..71
Daniel J. Bernstein, Detecting Perfect Powers in Essentially Linear Time, Mathematics of Computation. Vol. 67, 233, 1253-1283 (1998).
Marco Ripà, On some conjectures concerning perfect powers, ResearchGate (2022).

Cf. A001292, A058183, A062503, A134804, A181129, A352329, A352991.

z = 1; Do[r = Range[k];
n = ToExpression[StringJoin[ToString[#] & /@ r]];
If[And[Mod[n, 9] != 3, Mod[n, 9] != 6], d = DigitCount[n];
  s = IntegerPart[Sqrt[10^(IntegerLength[n] - 1)]];
  f = IntegerPart[Sqrt[10^(IntegerLength[n])]];
  Do[y = x^2;
   If[DigitCount[y] == d, c = True;
    Do[If[Not[StringContainsQ[ToString[y], ToString[i]]],
      c = False], {i, 10, k}]; If[c, Print[z, " ", y]; z++]], {x, s,
    f}]], {k, 1, 10}] (* Aldo Roberto Pessolano, May 12 2022 *)

Digit sum of a(n) is always congruent to 0 or 1 modulo 9.

a(n) = m^2, where the integer m := m(n) is not a perfect power itself (conjectured).

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.

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