login
A318763
Numbers k such that neighboring digits of k^22 are distinct.
1
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
1,2
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
EXAMPLE
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.
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 *)
CROSSREFS
Cf. A090516.
Sequence in context: A107467 A191425 A155881 * A041731 A041331 A290033
KEYWORD
nonn,base
AUTHOR
Robert Israel, 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
STATUS
approved