The OEIS mourns the passing of Jim Simons and is grateful to the Simons Foundation for its support of research in many branches of science, including the OEIS.
login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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
Sequence in context: A184772 A015425 A352329 * A345609 A346283 A205026
KEYWORD
nonn,base
AUTHOR
Marco Ripà, Apr 17 2022
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified June 12 16:17 EDT 2024. Contains 373334 sequences. (Running on oeis4.)