A090516 -id:A090516 - cp's OEIS frontend

A075309 Distinct-digit perfect powers.

1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81, 125, 128, 169, 196, 216, 243, 256, 289, 324, 361, 512, 529, 576, 625, 729, 784, 841, 961, 1024, 1089, 1296, 1369, 1728, 1764, 1849, 1936, 2048, 2187, 2197, 2304, 2401, 2601, 2704, 2809, 2916, 3025, 3125, 3249, 3481

Offset: 1

Zak Seidov, Oct 11 2002

Of 1110 perfect powers < 1000000, 259 are distinct-digit.
Sequence is finite. What is the index of the last term? Note that 2^30 = 1073741824, hence the highest power that occurs < 30. The frequency chart of a power r, 2 < r < 30 may be of some interest and could be included. - Amarnath Murthy, Dec 06 2003
There are a total of 657 distinct terms, the last of which is 99066^2=9814072356. The highest power occurs in 2^29. There are 609 squares, 39 cubes, 19 fourth powers, 9 fifth powers, 4 sixth powers, 4 seventh powers, 3 eighth powers, 2 ninth powers, 2 tenth powers and one each of powers 11, 12, 13, 14, 15, 20 and 29. These counts to not add to 657 because 1 is not counted and some powers, such as 2^4 = 4^2 = 16, are counted twice. - T. D. Noe, Aug 09 2005

			100,121,144,343 etc. are not members.

T. D. Noe, Table of n, a(n) for n = 1..657 (full sequence)

Cf. A001597, A090516.

lim:=floor(sqrt(9876543210)): A075309:={1}: for n from 2 to lim do k:=2: p:=n^k: while p<=9876543210 do p:=n^k: pandig:=true: d:=convert(p, base, 10): for j from 0 to 9 do if(numboccur(j, d)>1)then pandig:=false: break: fi: od: if(pandig)then A075309 := A075309 union {p}: fi: k:=k+1: od: od: op(sort(convert(A075309, list))); # Nathaniel Johnston, Jun 23 2011

lst={1}; Do[k=1; While[k++; n=k^pow; n<10^10, d=IntegerDigits[n]; If[Length[Union[d]]==Length[d], AppendTo[lst, n]]], {pow, 2, 29}]; lst=Union[lst] (* T. D. Noe *)

More terms from David Wasserman, Jan 16 2005

A318763 Numbers k such that neighboring digits of k^22 are distinct.

Original entry on oeis.org

1, 2, 3, 5, 8, 26, 27, 41, 83, 2102, 6972, 7761, 8942, 20003, 59305, 75663, 123623, 185746, 254268, 394334, 10655255, 14778909, 93300158, 565765348, 768508444, 860579661, 1871501781, 2081745983, 5414493192, 6055361574, 14177422842, 19061134031, 36554529447

Offset: 1

Robert Israel, Sep 03 2018

Heuristically, since k^22 has approximately 22*log_10(k) digits, the probability its neighboring digits are all distinct is approximately (9/10)^(22*log_10(k)) = k^(-22 log_10(10/9)). Since 22*log_10(10/9) = 1.006664792... > 1, we should expect this sequence to be finite.
a(27) if it exists is greater than 10^9. - Robert Price, Sep 06 2018
The number of d-digit numbers that are 22nd powers is approximately N(d) = 10^(d/22) - 10^((d-1)/22). If, as a fairly simple heuristic approach, we consider each d-digit 22nd power m as having a probability of (9/10)^(d-1) of having no runs of two or more of the same digit (so that m^(1/22) is a term of this sequence), then the expected number of such d-digit 22nd powers is about (9/10)^(d-1)*N(d) = (9/10)^(d-1)*(10^(d/22) - 10^((d-1)/22)), so the expected number of j-digit terms in this sequence should be about Sum_{d=22*(j-1)+1..22*j} (9/10)^(d-1)*(10^(d/22) - 10^((d-1)/22)); e.g., for j = 11..15, the expected numbers of j-digit terms in this sequence would be about 2.0669, 2.0354, 2.0044, 1.9739, and 1.9438, respectively. Perhaps surprisingly, this heuristic would indicate that this sequence should include about 136 terms beyond 10^10, and that the final term in this sequence -- not its 22nd power, but the term itself -- is most likely a number between 300 and 400 digits long. - Jon E. Schoenfield, Sep 07 2018

			a(3) = 3 is a member because 3^22 = 31381059609 has all its neighboring digits distinct. 4 is not a member because 4^22 = 17592186044416 contains 44.

Cf. A090516.

filter:= proc(n) local L;
  L:= convert(n,base,10);
  not member(0, L[2..-1]-L[1..-2])
end proc:
select(t -> filter(t^22), [$1..10^6]);

Select[Range[10^6], 0 != Times @@ Differences[IntegerDigits[#^22]] &] (* Giovanni Resta, Sep 03 2018 *)

a(21)-a(26) from Robert Price, Sep 06 2018

a(27)-a(33) from Jon E. Schoenfield, Sep 10 2018

A175031 Sequence of increasing positive perfect powers such that neighboring digits in the sequence are distinct.

Original entry on oeis.org

1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81, 216, 243, 256, 289, 324, 343, 484, 512, 529, 576, 729, 784, 841, 961, 2025, 2048, 2187, 2197, 2304, 2401, 2601, 2704, 2809, 2916, 3025, 3125, 3136, 3249, 3481, 3721, 3969, 4096, 4356, 4624, 5041, 5184, 5329, 5476

Offset: 1

Zak Seidov, Nov 06 2009

Subsequence of A090516 (Perfect powers (at least a square) in which neighboring digits are distinct.) which itself is subsequence of A001597 ( Perfect powers: m^k where m is an integer and k >= 2) which itself is Subsequence of A000027 (natural numbers).

m=26000;a=1;A001597=Union[Join[{a},Flatten[Table[n^i,{n,2,Sqrt[m]},{i,2,Log[n,m]}]]]]; A090516=Select[A001597,Max[Length/@Split[IntegerDigits[ # ]]]==1&]; S={1};ia1={a};Do[si=A090516[[i]];If[IntegerDigits[si][[1]]!=ia1,AppendTo[S,si];ia1=IntegerDigits[si][[ -1]]],{i,2,Length[se1]}]; S

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A075309 Distinct-digit perfect powers.

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A318763 Numbers k such that neighboring digits of k^22 are distinct.

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Maple

Mathematica

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A175031 Sequence of increasing positive perfect powers such that neighboring digits in the sequence are distinct.

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Mathematica