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A190300 Composite numbers that are not Brazilian. 7
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 (list; graph; refs; listen; history; text; internal format)
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

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

Intersection of A002808 and A220570.

Intersection of A001358 and A220570.

Sequence in context: A246569 A326063 A085721 * A081614 A192220 A215477

Adjacent sequences:  A190297 A190298 A190299 * A190301 A190302 A190303

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

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Last modified January 22 10:52 EST 2020. Contains 331144 sequences. (Running on oeis4.)