A190300 Composite numbers that are not Brazilian.

4, 6, 9, 25, 49, 169, 289, 361, 529, 841, 961, 1369, 1681, 1849, 2209, 2809, 3481, 3721, 4489, 5041, 5329, 6241, 6889, 7921, 9409, 10201, 10609, 11449, 11881, 12769, 16129, 17161, 18769, 19321, 22201, 22801, 24649, 26569, 27889, 29929, 32041, 32761, 36481, 37249, 38809, 39601, 44521, 49729

Offset: 1

Author

N. J. A. Sloane, May 14 2011

Keywords

• nonn

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

Examples

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

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.

Crossrefs

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

Intersection of A002808 and A220570.

Intersection of A001358 and A220570.

Programs

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 *)

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

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

Extensions

a(6)-a(24) from Nathaniel Johnston, May 24 2011

a(25) onward from Robert G. Wilson v, Mar 30 2014