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A290015 Brazilian numbers which have exactly two Brazilian representations. 3

%I

%S 15,18,21,26,28,30,31,32,44,45,50,52,56,57,62,64,68,75,76,85,86,91,92,

%T 93,98,99,110,111,116,117,129,133,146,147,148,153,164,175,183,188,207,

%U 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

%N Brazilian numbers which have exactly two Brazilian representations.

%C These numbers could be called 2-Brazilian numbers.

%C 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.

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

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Goormaghtigh_conjecture">Goormaghtigh conjecture</a>

%e 18 = 2 * 9 = 22_8 = 3 * 6 = 33_5.

%e 26 = 2 * 13 = 2 * 111_3 = 222_3 = 22_12.

%e 31 = 11111_2 = 111_5;

%e 8191 = 1111111111111_2 = 111_90.

%p bresilienbaseb:=proc(n,b)

%p local r,q,coupleq:

%p if n<b then

%p return [1,n]

%p else

%p r:=(n mod b):

%p q:=(n-r)/b

%p coupleq:=bresilienbase(q,b):

%p if r=coupleq[2] and r>0 then

%p return [couple[1]+1,r]

%p else

%p return [0,0]

%p end if

%p end if

%p end proc;

%p bresil:=proc(n)

%p local b,L,k,t:

%p k:=0:

%p for b from 2 to (n-2) do

%p t:=bresilienbase(n,b):

%p if t[1]>0 then

%p k:=k+1

%p L[k]:=[b,t[1],t[2]]:

%p end if:

%p end do:

%p seq(L[i],i=1..k);

%p end proc;

%p nbbresil:=n->nops([bresil(n)]);

%p #Numbers 2 times Brazilian

%p for n from 1 to 100 do if nbbresil(n)=2 then print(n,bresil(n)) else fi; od:

%t 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 *)

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

%K nonn,base

%O 1,1

%A _Bernard Schott_, Jul 17 2017

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Last modified February 20 19:02 EST 2020. Contains 332082 sequences. (Running on oeis4.)