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, 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

Offset: 1

Author

V. Raman, Sep 07 2012

Keywords

• sign
• base

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.