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%I #87 May 13 2022 14:04:05
%S 1,13527684,34857216,65318724,73256481,81432576,139854276,152843769,
%T 157326849,215384976,245893761,254817369,326597184,361874529,
%U 375468129,382945761,385297641,412739856,523814769,529874361,537219684,549386721,587432169,589324176,597362481,615387249,627953481,653927184
%N Terms of A352991 which are perfect powers.
%C It appears that all terms are terms of A062503.
%C 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).
%C 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 Therefore, all terms are necessarily congruent modulo 9 to 0 or 1 (see Marco Ripà link).
%C 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
%H Aldo Roberto Pessolano, <a href="/A353025/b353025.txt">Table of n, a(n) for n = 1..71</a>
%H Daniel J. Bernstein, <a href="https://doi.org/10.1090/S0025-5718-98-00952-1">Detecting Perfect Powers in Essentially Linear Time</a>, Mathematics of Computation. Vol. 67, 233, 1253-1283 (1998).
%H Marco Ripà, <a href="https://doi.org/10.13140/RG.2.2.30313.77921/1">On some conjectures concerning perfect powers</a>, ResearchGate (2022).
%F Digit sum of a(n) is always congruent to 0 or 1 modulo 9.
%F a(n) = m^2, where the integer m := m(n) is not a perfect power itself (conjectured).
%e 75910168324 is a term since 75910168324 = 275518^2.
%t z = 1; Do[r = Range[k];
%t n = ToExpression[StringJoin[ToString[#] & /@ r]];
%t If[And[Mod[n, 9] != 3, Mod[n, 9] != 6], d = DigitCount[n];
%t s = IntegerPart[Sqrt[10^(IntegerLength[n] - 1)]];
%t f = IntegerPart[Sqrt[10^(IntegerLength[n])]];
%t Do[y = x^2;
%t If[DigitCount[y] == d, c = True;
%t Do[If[Not[StringContainsQ[ToString[y], ToString[i]]],
%t c = False], {i, 10, k}]; If[c, Print[z, " ", y]; z++]], {x, s,
%t f}]], {k, 1, 10}] (* _Aldo Roberto Pessolano_, May 12 2022 *)
%Y Cf. A001292, A058183, A062503, A134804, A181129, A352329, A352991.
%K nonn,base
%O 1,2
%A _Marco Ripà_, Apr 17 2022
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