

A216485


a(n) is the least value of k such that k*n uses only the digit 2, or a(n) = 1 if no such multiple exists.


2



2, 1, 74, 1, 1, 37, 31746, 1, 24691358, 1, 2, 1, 17094, 15873, 1, 1, 130718954248366, 12345679, 11695906432748538, 1, 10582, 1, 96618357487922705314, 1, 1, 8547, 8230452674897119341563786, 1, 76628352490421455938697318, 1, 7168458781362, 1, 6734, 65359477124183, 1, 1, 6, 5847953216374269, 5698, 1, 542, 5291, 5167958656330749354
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OFFSET

1,1


COMMENTS

a(n) <= 2(10^n 1)/(9n). a(n) = 1 if and only if n is a multiple of 4 or 5. If n is a multiple of 4 then a(n) = 1 since 222....222 is not a multiple of 4. If n is a multiple of 5 then all multiples of n ends with the digit 0 or 5 and a(n) = 1. If n is odd and not a multiple of 4 or 5, then by the pigeonhole principle, two different repunits will have the same remainder modulo n. Their difference will be of the form 11...1110..0 which is a multiple of n. Since n and 10 are coprime, n is a divisor of a repunit and a(n) != 1. If n is even and not a multiple of 4 or 5, we take n/2 and use the same argument to show that n/2 is a divisor of a repunit and a(n) != 1.  Chai Wah Wu, Jun 21 2015


LINKS

Chai Wah Wu, Table of n, a(n) for n = 1..1000


CROSSREFS

Cf. A004290, A079339, A181060, A181061.
Sequence in context: A104024 A233472 A284596 * A096681 A247793 A067276
Adjacent sequences: A216482 A216483 A216484 * A216486 A216487 A216488


KEYWORD

sign,base


AUTHOR

V. Raman, Sep 07 2012


STATUS

approved



