A090516 Perfect powers (at least a square) in which neighboring digits are distinct.

1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81, 121, 125, 128, 169, 196, 216, 243, 256, 289, 324, 343, 361, 484, 512, 529, 576, 625, 676, 729, 784, 841, 961, 1024, 1089, 1296, 1369, 1521, 1681, 1728, 1764, 1849, 1936, 2025, 2048, 2187, 2197, 2304, 2401, 2601

Offset: 1

Author

Amarnath Murthy, Dec 06 2003

Keywords

• base
• easy
• nonn

Comments

Sequence must be infinite but a proof is needed. Subsidiary sequences; Perfect squares or perfect cubes etc. in which neighboring digits are distinct.

On the other hand, for k >= 22 we might expect only finitely many k-th powers where neighboring digits are distinct (see A318763). - Robert Israel, Sep 03 2018

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001597 (perfect powers), A075309 (distinct-digit perfect powers), A318763.

Programs

Maple

N:= 5000:
filter:= proc(n) local L;
  L:= convert(n,base,10);
  not member(0, L[2..-1]-L[1..-2])
end proc:
P:= sort(convert({seq(seq(i^k,i=1..floor(N^(1/k))),k=2..ilog2(N))},list)):
select(filter, P); # Robert Israel, Sep 03 2018

Extensions

Corrected and extended by Rick L. Shepherd, Jul 01 2005

Offset corrected by Robert Israel, Sep 03 2018