A353025 Terms of A352991 which are perfect powers.

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

Author

Marco Ripà, Apr 17 2022

Keywords

• nonn
• base

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

Examples

			75910168324 is a term since 75910168324 = 275518^2.
		

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).

Crossrefs

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

Programs

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 *)

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).