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

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

Author

Robert Israel, Sep 03 2018

Keywords

• nonn
• base

Comments

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

Examples

			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.
		

Crossrefs

Cf. A090516.

Programs

Maple

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]);

Mathematica

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

Extensions

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

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