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A190300 Composite numbers that are not Brazilian. 8

%I

%S 4,6,9,25,49,169,289,361,529,841,961,1369,1681,1849,2209,2809,3481,

%T 3721,4489,5041,5329,6241,6889,7921,9409,10201,10609,11449,11881,

%U 12769,16129,17161,18769,19321,22201,22801,24649,26569,27889,29929,32041,32761,36481,37249,38809,39601,44521,49729

%N Composite numbers that are not Brazilian.

%C Other than the term 6 and the missing term 121, is this sequence the same as A001248? - _Nathaniel Johnston_, May 24 2011

%C From _Bernard Schott_, Dec 04 2012: (Start)

%C Yes, because

%C 1) 4 is not a Brazilian number [4 = 100_2].

%C 2) 6 is not a Brazilian number [6 = 110_2 = 20_3 = 12_4].

%C 3) Theorem 1, page 32 of Quadrature article mentioned in links: If n > 7 is not Brazilian, then n is a prime or the square of a prime.

%C 4) Theorem 5, page 37 of Quadrature article mentioned in links: The only square of prime number which is Brazilian is 121 = 11^2 = 11111_3.

%C (End)

%C There is an infinity of composite numbers that are not Brazilian: Corollary 2, page 37 of Quadrature article in links (consider the sequence of squares of prime numbers for p >= 13). - _Bernard Schott_, Dec 17 2012

%C Also semiprimes that are not Brazilian. - _Bernard Schott_, Apr 11 2019

%H Robert Israel, <a href="/A190300/b190300.txt">Table of n, a(n) for n = 1..10000</a> (first 208 terms from Robert G. Wilson v)

%H Bernard Schott, <a href="/A125134/a125134.pdf">Les nombres brésiliens</a>, Quadrature, no. 76, avril-juin 2010, pages 30-38; included here with permission from the editors of Quadrature.

%F a(1) = 2^2 = p_1^2, a(2) = 2*3 = p_1*p_2, a(3) = 3^2 = p_2^2, a(4) = 5^2 = p_3^2, a(5) = 7^2 = p_4^2, a(6) = 13^2 = p_6^2, ..., for n >= 6, a(n) = p_n^2, where p_k is the k-th prime number. - _Bernard Schott_, Dec 04 2012

%e a(10) = p_10^2 = 29^2 = 841.

%p 4, 6, 9, 25, 49,seq(ithprime(i)^2, i=6..100); # _Robert Israel_, Apr 17 2019

%t brazBases[n_] := Select[Range[2, n - 2], Length[Union[IntegerDigits[n, #]]] == 1 &]; Select[Range[2, 10000], ! PrimeQ[#] && brazBases[#] == {} &] (* _T. D. Noe_, Dec 26 2012 *)

%t f[n_] := Block[{b = 2}, While[ Length@ Union@ IntegerDigits[n, b] != 1, b++]; b]; k = 4; lst = {}; While[k < 50001, If[ !PrimeQ@ k && 1 + f@ k == k, AppendTo[lst, k]]; k++]; lst (* _Robert G. Wilson v_, Mar 30 2014 *)

%o (PARI) isnotb(n) = my(c=0, d); for(b=2, n-2, d=digits(n, b); if(vecmin(d)==vecmax(d), c=n; break); c++); (c==max(n-3, 0)); \\ A220570

%o lista(nn) = forcomposite(n=1, nn, if (isnotb(n), print1(n, ", "))); \\ _Michel Marcus_, Apr 14 2019

%Y Cf. A085104, A125134, A189891, A220571, A307507.

%Y Intersection of A002808 and A220570.

%Y Intersection of A001358 and A220570.

%K nonn

%O 1,1

%A _N. J. A. Sloane_, May 14 2011

%E a(6)-a(24) from _Nathaniel Johnston_, May 24 2011

%E a(25) onward from _Robert G. Wilson v_, Mar 30 2014

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Last modified February 23 16:17 EST 2020. Contains 332175 sequences. (Running on oeis4.)