A167783 Numbers that are repdigits with length > 2 in more than one base.

31, 63, 255, 273, 364, 511, 546, 728, 777, 931, 1023, 1365, 1464, 2730, 3280, 3549, 3783, 3906, 4095, 4557, 6560, 7566, 7812, 8191, 9114, 9331, 9841, 10507, 11349, 11718, 13671, 14043, 14763, 15132, 15624, 16383, 18291, 18662, 18915, 19608, 19682, 21845, 22351, 22698

Offset: 1

Author

Andrew Weimholt, Nov 12 2009

Keywords

• nonn
• base

Comments

Definition requires "length > 2" because all numbers n > 2 are trivially represented as "11" in base n-1.

From Daniel Forgues, Nov 13 2009: (Start)

0 = 00 = 000 = 0000 = 00000 = 000000 = 0000000 = 00000000 = ... in any positional number representation (includes fixed base radix b > 1, mixed base radix with each b_i > 1, i >= 0, such as factorial and primorial based radix...)

The sequence definition should be read as:

Nonnegative integers that are repdigits with length > 2 in more than one fixed base radix b > 1.

Considering all fixed and mixed base radix would include many more nonnegative integers (but not the integers 1 to 6) which are repdigits with length > 2 in more than one radix. (End)

From Bernard Schott, Aug 08 2017: (Start)

In this sequence data, the first number which is repdigit, with length > 2, in more than two bases is the twelfth Mersenne number 4095 with four Brazilian representations: M_12 = 4095 = 111111111111_2 = 333333_4 = 7777_8 = (15 15 15)_16.

The Mersenne number M_15 is the first number which is repdigit in exactly three bases with M_15 = 32767 = 111111111111111_2 = 77777_8 = (31 31 31)_32.

Only two numbers are repunits in more than one base: the Mersenne primes 31 and 8191 (Examples and A119598).

Some numbers are once repunit and once multiple of a Brazilian prime such that Mersenne number M_9 = 511 = 7 * 73 = 111111111_2 = 7 * 111_8 = 777_8.

Some numbers are once repunit and once multiple of a composite repunit such that Mersenne number M_6 = 63 = 3 * 21 = 111111_2 = 3 * 111_4 = 333_4.

Some numbers are repdigits in two different bases: 546 = 666_9 = 222_16. (End)

Examples

			31 is in the list because 31 = 11111_2 = 111_5;
8191 = 1111111111111_2 = 111_90;
10507 = {19 19 19}_23 = 111_102.
		

Links

• David Trimas, Table of n, a(n) for n = 1..8424 (first 158 terms from Michel Marcus)
• David Trimas, Wolfram Cloud Implementation of A167783
• Wolfram Demonstrations Project, Mixed Radix Number Representations

Crossrefs

Cf. A167782 (numbers that are repdigits with length > 2 in some base).

Cf. A010785 (repdigits (base 10)).

Cf. A053696 (numbers which are repunits in some base).

Cf. A158235 (numbers n whose square is a repdigit in some base < n).

Cf. A290869 (Numbers that are repdigits with length > 2 in more than two bases).

Programs

Mathematica

Select[Range[550], Function[n, 1 < Count[Range[2, n - 1], ?(And[Length@ DeleteCases[#, 0] == 1, Union[#][[2]] > 2] &@ DigitCount[n, #] &)]]] (* _Michael De Vlieger, Aug 09 2017 *)

PARI

isok(n)=my(nb = 0); for (b=2, n-1, d = digits(n, b); if ((#d > 2) && (#Set(d) == 1), nb++); if (nb > 1, return (1));); return (0); \\ Michel Marcus, Aug 08 2017

Extensions

a(41)-a(44) from Bernard Schott, Aug 08 2017