A341057 Numbers without Brazilian divisors.

1, 2, 3, 4, 5, 6, 9, 11, 17, 19, 23, 25, 29, 37, 41, 47, 53, 59, 61, 67, 71, 79, 83, 89, 97, 101, 103, 107, 109, 113, 131, 137, 139, 149, 151, 163, 167, 173, 179, 181, 191, 193, 197, 199, 223, 227, 229, 233, 239, 251, 257, 263, 269, 271, 277, 281, 283, 289, 293, 311

Offset: 1

Author

Bernard Schott, Feb 04 2021

Keywords

• nonn
• base

Comments

The first 16 terms are the first 16 terms of A220570 (non-Brazilian numbers), then a(17) = 53 while A220570(17) = 49.

m is a term iff m = 1, or m = 6, or m is a non-Brazilian prime (A220627) or m is the square of a non-Brazilian prime, except for 121 that is Brazilian (see examples).

Examples

			One example for each type of terms that has k divisors:
-> k=1: 1 is the smallest number not Brazilian, hence 1 is the first term.
-> k=2: 17 is a prime non-Brazilian, hence 17 is a term.
-> k=3: 25 has three divisors {1, 5, 25} that are all not Brazilian, hence 25 is another term.
-> k=4: 6 has four divisors {1, 2, 3, 6} that are all not Brazilian, hence 6 is the term that has the largest number of divisors.
		

Crossrefs

Cf. A125134, A340795, A308851, A341058 (with 1 Brazilian divisor).

Subsequence of A220570 (non-Brazilian numbers).

Supersequence of A220627 (non-Brazilian primes).

Programs

Mathematica

brazQ[n_] := Module[{b = 2, found = False}, While[b < n - 1 && Length @ Union @ IntegerDigits[n, b] > 1, b++]; b < n - 1]; q[n_] := AllTrue[Divisors[n], ! brazQ[#] &]; Select[Range[300], q] (* Amiram Eldar, Feb 04 2021 *)

PARI

isb(n) = for(b=2, n-2, my(d=digits(n, b)); if(vecmin(d)==vecmax(d), return(1))); \\ A125134
isok(n) = fordiv(n, d, if (isb(d), return(0))); return(1); \\ Michel Marcus, Feb 07 2021

Formula

A340795(a(n)) = 0.