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A290015 Brazilian numbers which have exactly two Brazilian representations. 3
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 (list; graph; refs; listen; history; text; internal format)
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 of course highly Brazilian in A066044.

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. A066044, A085104, A125134, A220570, A220571, A257521, A284758, A288783, A290016, A290017, A290018.

Sequence in context: A063781 A125006 A125008 * A274979 A083823 A222682

Adjacent sequences:  A290012 A290013 A290014 * A290016 A290017 A290018

KEYWORD

nonn,base

AUTHOR

Bernard Schott, Jul 17 2017

STATUS

approved

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Last modified January 22 10:25 EST 2020. Contains 331144 sequences. (Running on oeis4.)