A290015 Brazilian numbers which have exactly two Brazilian representations.

15, 18, 21, 26, 28, 30, 31, 32, 44, 45, 50, 52, 56, 57, 62, 64, 68, 75, 76, 85, 86, 91, 92, 93, 98, 99, 110, 111, 116, 117, 129, 133, 146, 147, 148, 153, 164, 175, 183, 188, 207, 212, 215, 219, 236, 243, 244, 245, 259, 261, 268, 275, 279, 284, 314, 316, 325, 332, 338, 341, 343, 356, 363, 365, 369, 381, 387, 388

Offset

1,1

Comments

These numbers could be called 2-Brazilian numbers.

The smallest number of this sequence is 15 which is also the smallest odd composite Brazilian in A257521 with 15 = 11111_2 = 33_4. The number 15 is highly Brazilian in A329383.

Following the Goormaghtigh conjecture, only two primes, 31 and 8191, which are both Mersenne numbers, are Brazilian in two different bases (A119598).

Links

Table of n, a(n) for n=1..68.

Wikipedia, Goormaghtigh conjecture

Example

18 = 2 * 9 = 22_8 = 3 * 6 = 33_5.
26 = 2 * 13 = 2 * 111_3 = 222_3 = 22_12.
31 = 11111_2 = 111_5;
8191 = 1111111111111_2 = 111_90.

Maple

bresilienbaseb:=proc(n, b)
local r, q, coupleq:
if n<b then
return [1, n]
else
r:=(n mod b):
q:=(n-r)/b
coupleq:=bresilienbase(q, b):
if r=coupleq[2] and r>0 then
return [couple[1]+1, r]
else
return [0, 0]
end if
end if
end proc;
bresil:=proc(n)
local b, L, k, t:
k:=0:
for b from 2 to (n-2) do
t:=bresilienbase(n, b):
if t[1]>0 then
k:=k+1
L[k]:=[b, t[1], t[2]]:
end if:
end do:
seq(L[i], i=1..k);
end proc;
nbbresil:=n->nops([bresil(n)]);
#Numbers 2 times Brazilian
for n from 1 to 100 do if nbbresil(n)=2 then print(n, bresil(n)) else fi; od:

Mathematica

Flatten@ Position[#, 2] &@ Table[Count[Range[2, n - 2], _?(And[Length@ # != 1, Length@ Union@ # == 1] &@ IntegerDigits[n, #] &)], {n, 400}] (* Michael De Vlieger, Jul 18 2017 *)

Crossrefs

Cf. A085104, A125134, A220570, A220571, A257521, A284758, A288783, A290016, A290017, A290018, A329383.

Sequence in context: A063779 A125006 A125008 * A354818 A274979 A083823

Adjacent sequences: A290012 A290013 A290014 * A290016 A290017 A290018

Keyword

nonn,base

Author

Bernard Schott, Jul 17 2017

Status

approved