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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 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

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Last modified December 8 20:37 EST 2021. Contains 349596 sequences. (Running on oeis4.)