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A353025 Terms of A352991 which are perfect powers. 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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

LINKS

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).

FORMULA

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).

EXAMPLE

75910168324 is a term since 75910168324 = 275518^2.

MATHEMATICA

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 *)

CROSSREFS

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

Sequence in context: A184772 A015425 A352329 * A345609 A346283 A205026

Adjacent sequences:  A353022 A353023 A353024 * A353026 A353027 A353028

KEYWORD

nonn,base

AUTHOR

Marco Ripà, Apr 17 2022

STATUS

approved

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Last modified August 14 16:46 EDT 2022. Contains 356122 sequences. (Running on oeis4.)