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A306540 Least k >= 1 such that the decimal expansion of n^k starts with 99 or 100. 1
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 (list; graph; refs; listen; history; text; internal format)
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
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
KEYWORD
nonn,base,look
AUTHOR
Robert Israel, Feb 22 2019
STATUS
approved

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Last modified March 2 12:34 EST 2024. Contains 370467 sequences. (Running on oeis4.)