A308851 Numbers >= 2 all of whose divisors > 1 are Brazilian.

7, 13, 31, 43, 73, 91, 127, 157, 211, 217, 241, 301, 307, 403, 421, 463, 511, 559, 601, 757, 889, 949, 1093, 1099, 1123, 1333, 1477, 1483, 1651, 1687, 1723, 2041, 2149, 2263, 2551, 2743, 2801, 2821, 2947, 2971, 3133, 3139, 3241, 3307, 3541, 3907, 3913, 3937

Offset: 1

Author

Bernard Schott, Jun 28 2019

Keywords

• nonn

Comments

The terms of this sequence are the Brazilian primes and the products of two or more distinct Brazilian primes.

There are no even numbers because 2 is not Brazilian.

Examples

			91 is a term because all divisors of 91 that are > 1: {7, 13, 91} are Brazilian numbers with 7 = 111_2, 13 = 111_3 and 91 = 77_12.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..1000
• Index entries for sequences related to Brazilian numbers

Crossrefs

Cf. A085104 (subsequence), A125134.

Similar with even numbers: A000079, with odd numbers: A005408, with palindromes: A062687, with repdigits: A190217.

Programs

Mathematica

brazQ[n_] := Block[{k, b, ok}, If[FindInstance[k (1 + b) == n && 1 < b < n - 1 && 0 < k < b, {k, b}, Integers] != {}, True, b = 2; ok = False; While[1 + b + b^2 <= n && ! ok, ok = Length@ Union@ IntegerDigits[n, b++] == 1]; ok]]; Select[ Range[3, 4000, 2], AllTrue[ Rest@ Divisors@ #, brazQ] &] (* Giovanni Resta, Jun 29 2019 *)
max = 5000; fQ[n_] := Module[{b = 2, found = False}, While[b < n - 1 && Length[Union[IntegerDigits[n, b]]] > 1, b++]; b < n - 1]; A125134 = Select[Range[4, max], fQ]; Select[Range[2, max], Intersection[A125134, Rest[Divisors[#]]] == Rest[Divisors[#]] &] (* Vaclav Kotesovec, Jun 29 2019, using a subroutine from T. D. Noe *)

PARI

isb(n) = for(b=2, n-2, d=digits(n, b); if(vecmin(d)==vecmax(d), return(1)));
isok(n) = {fordiv(n, d, if ((d>1) && ! isb(d), return (0));); return (1);} \\ Michel Marcus, Jun 29 2019