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

Author

Zak Seidov, Oct 11 2002

Keywords

• easy
• nonn
• base
• fini
• full

Comments

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

Examples

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

Links

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

Crossrefs

Cf. A001597, A090516.

Programs

Maple

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

Mathematica

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 *)

Extensions

More terms from David Wasserman, Jan 16 2005