A306540 Least k >= 1 such that the decimal expansion of n^k starts with 99 or 100.

1, 93, 153, 98, 93, 9, 71, 31, 153, 1, 145, 101, 79, 130, 159, 49, 13, 47, 61, 93, 90, 73, 47, 192, 98, 147, 51, 123, 93, 153, 116, 97, 27, 143, 68, 151, 44, 188, 22, 98, 31, 69, 30, 115, 124, 86, 61, 160, 71, 93, 106, 81, 29, 71, 104, 139, 127, 93, 48, 9, 177, 53, 5, 129, 155, 133, 23, 197, 211

Offset

1,2

Comments

Digits after the decimal point are allowed, so a(1)=a(10)=1.

a(n) always exists and is bounded: see proof at A217157.

Conjecture: a(n) <= 231.

Links

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

Example

a(6) = 9 because 6^9 = 10077696 and no lower power of 6 starts with 99 or 100.

Maple

f:=proc(n) local k, v, r;
  v:= 1;
  for k from 1 do
    v:= v*n;
    r:= v/10^(ilog10(v));
    if r < 101/100 or r >= 99/10 then return k fi
  od
end proc:
map(f, [$1..100]);

Prog

(Python)
def A306540(n):
    if n == 1 or n == 10:
        return 1
    k, nk = 1, n
    while True:
        s = str(nk)
        if s[:2] == '99' or s[:3] == '100':
            return k
        k += 1
        nk *= n # Chai Wah Wu, Feb 22 2019

Crossrefs

Cf. A217157.

Sequence in context: A255991 A213114 A250366 * A044425 A044806 A043363

Adjacent sequences: A306537 A306538 A306539 * A306541 A306542 A306543

Keyword

nonn,base,look

Author

Robert Israel, Feb 22 2019

Status

approved