OFFSET
1,1
COMMENTS
Other than the term 6 and the missing term 121, is this sequence the same as A001248? - Nathaniel Johnston, May 24 2011
From Bernard Schott, Dec 04 2012: (Start)
Yes, because
1) 4 is not a Brazilian number [4 = 100_2].
2) 6 is not a Brazilian number [6 = 110_2 = 20_3 = 12_4].
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.
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.
(End)
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
Also semiprimes that are not Brazilian. - Bernard Schott, Apr 11 2019
LINKS
Robert Israel, Table of n, a(n) for n = 1..10000 (first 208 terms from Robert G. Wilson v)
Bernard Schott, Les nombres brésiliens, Quadrature, no. 76, avril-juin 2010, pages 30-38; included here with permission from the editors of Quadrature.
FORMULA
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
EXAMPLE
a(10) = p_10^2 = 29^2 = 841.
MAPLE
4, 6, 9, 25, 49, seq(ithprime(i)^2, i=6..100); # Robert Israel, Apr 17 2019
MATHEMATICA
brazBases[n_] := Select[Range[2, n - 2], Length[Union[IntegerDigits[n, #]]] == 1 &]; Select[Range[2, 10000], ! PrimeQ[#] && brazBases[#] == {} &] (* T. D. Noe, Dec 26 2012 *)
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 *)
PROG
(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
lista(nn) = forcomposite(n=1, nn, if (isnotb(n), print1(n, ", "))); \\ Michel Marcus, Apr 14 2019
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, May 14 2011
EXTENSIONS
a(6)-a(24) from Nathaniel Johnston, May 24 2011
a(25) onward from Robert G. Wilson v, Mar 30 2014
STATUS
approved