cp's OEIS Frontend

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

%I A167783 #69 Jun 25 2025 12:07:22
%S A167783 31,63,255,273,364,511,546,728,777,931,1023,1365,1464,2730,3280,3549,
%T A167783 3783,3906,4095,4557,6560,7566,7812,8191,9114,9331,9841,10507,11349,
%U A167783 11718,13671,14043,14763,15132,15624,16383,18291,18662,18915,19608,19682,21845,22351,22698
%N A167783 Numbers that are repdigits with length > 2 in more than one base.
%C A167783 Definition requires "length > 2" because all numbers n > 2 are trivially represented as "11" in base n-1.
%C A167783 From _Daniel Forgues_, Nov 13 2009: (Start)
%C A167783 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...)
%C A167783 The sequence definition should be read as:
%C A167783 Nonnegative integers that are repdigits with length > 2 in more than one fixed base radix b > 1.
%C A167783 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)
%C A167783 From _Bernard Schott_, Aug 08 2017: (Start)
%C A167783 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.
%C A167783 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.
%C A167783 Only two numbers are repunits in more than one base: the Mersenne primes 31 and 8191 (Examples and A119598).
%C A167783 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.
%C A167783 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.
%C A167783 Some numbers are repdigits in two different bases: 546 = 666_9 = 222_16. (End)
%H A167783 David Trimas, <a href="/A167783/b167783.txt">Table of n, a(n) for n = 1..8424</a> (first 158 terms from Michel Marcus)
%H A167783 David Trimas, <a href="https://www.wolframcloud.com/obj/dtrimas/Published/A167783.nb/">Wolfram Cloud Implementation of A167783</a>
%H A167783 Wolfram Demonstrations Project, <a href="http://demonstrations.wolfram.com/MixedRadixNumberRepresentations/">Mixed Radix Number Representations</a>
%e A167783 31 is in the list because 31 = 11111_2 = 111_5;
%e A167783 8191 = 1111111111111_2 = 111_90;
%e A167783 10507 = {19 19 19}_23 = 111_102.
%t A167783 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 *)
%o A167783 (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
%Y A167783 Cf. A167782 (numbers that are repdigits with length > 2 in some base).
%Y A167783 Cf. A010785 (repdigits (base 10)).
%Y A167783 Cf. A053696 (numbers which are repunits in some base).
%Y A167783 Cf. A158235 (numbers n whose square is a repdigit in some base < n).
%Y A167783 Cf. A290869 (Numbers that are repdigits with length > 2 in more than two bases).
%K A167783 nonn,base
%O A167783 1,1
%A A167783 _Andrew Weimholt_, Nov 12 2009
%E A167783 a(41)-a(44) from _Bernard Schott_, Aug 08 2017