A280660 Least k such that at least half of the last n digits of 2^k are 9.

12, 53, 53, 232, 93, 1862, 93, 3244, 93, 93, 93, 55754, 12864, 55756, 23353, 361353, 16441, 361353, 304362, 361353, 361353, 361353, 361353, 3748854, 3748854, 78055893, 66290232, 119133355, 119133355, 379371432, 20958353, 130883333, 20958353, 130883333

Offset

2,1

Comments

See the Mathematical Reflections link for a proof that a(n) exists for all n>1.

Links

Jon E. Schoenfield and Chai Wah Wu, Table of n, a(n) for n = 2..44 (terms for n = 2..42 from Jon E. Schoenfield)

Mathematical Reflections, Solution to Problem O316, Issue 6, 2014, p 26.

Jon E. Schoenfield, Magma program

Example

For n=2, k=12 with 2^k = 4096.

Maple

a:= proc(n) local m, t, k, c, h; m, t:= 10^n, 2048;
      for k from 12 do t:= 2*t mod m; h, c:= t, 0;
        while h>0 do if irem(h, 10, 'h')=9 then c:= c+2 fi od;
        if c >= n then return k fi
      od
    end:
seq(a(n), n=2..16);  # Alois P. Heinz, Jan 07 2017

Prog

(PARI) a(n) = my(k = 1, ok = 0); until (ok, vd = Vecrev(digits(2^k)); nb = sum(j=1, min(n, #vd), vd[j]==9); ok = (nb >= n/2); if (! ok, k++); ); k;
(Python)
def A280660(n):
    m, k, l = 10**n, 1, 2
    while True:
        if 2*str(l).count('9') >= n:
            return k
        k += 1
        l = (l*2) % m # Chai Wah Wu, Jan 07 2017

Crossrefs

Sequence in context: A195544 A248135 A307916 * A268186 A253130 A213549

Adjacent sequences: A280657 A280658 A280659 * A280661 A280662 A280663

Keyword

nonn,base

Author

Michel Marcus, Jan 07 2017

Extensions

a(17)-a(30) from Alois P. Heinz, Jan 07 2017

a(31)-a(35) from Jon E. Schoenfield, Jan 07 2017

Status

approved