A341058 -id:A341058 - cp's OEIS frontend

A340795 a(n) is the number of divisors of n that are Brazilian.

0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 1, 2, 1, 2, 0, 1, 0, 2, 2, 1, 0, 3, 0, 2, 1, 3, 0, 3, 1, 3, 1, 1, 2, 3, 0, 1, 2, 4, 0, 4, 1, 2, 2, 1, 0, 5, 1, 2, 1, 3, 0, 3, 1, 5, 1, 1, 0, 6, 0, 2, 3, 4, 2, 3, 0, 2, 1, 5, 0, 6, 1, 1, 2, 2, 2, 4, 0, 6, 2, 1, 0, 7, 1, 2, 1, 4, 0, 6

Offset: 1

Bernard Schott, Jan 21 2021

The cases a(n) = 0 and a(n) = 1 are respectively detailed in A341057 and A341058.

			For n = 16, the divisors are 1, 2, 4, 8 and 16. Only 8 = 22_3 and 16 = 22_7 are Brazilian numbers, so a(16) = 2.
For n = 30, the divisors are 1, 2, 3, 5, 6, 10, 15 and 30. Only 10 = 22_4, 15 = 33_4 and 30 = 33_9 are Brazilian numbers, so a(30) = 3.
For n = 49, the divisors are 1, 7 and 49. Only 7 = 111_2 is Brazilian, so a(49) = 1 although 49 that is square of prime <> 121 is not Brazilian.

David A. Corneth, Table of n, a(n) for n = 1..10000
Wikipédia, Nombre brésilien (in French).
Index entries for sequences related to Brazilian numbers.

Cf. A085104, A125134, A220627, A308851, A340796, A340797, A341057, A341058.

brazQ[n_] := Module[{b = 2, found = False}, While[b < n - 1 && Length[Union[IntegerDigits[n, b]]] > 1, b++]; b < n - 1]; a[n_] := DivisorSum[n, 1 &, brazQ[#] &]; Array[a, 100] (* Amiram Eldar, Jan 21 2021 *)

isb(n) = for(b=2, n-2, d=digits(n, b); if(vecmin(d)==vecmax(d), return(1))); \\ A125134
a(n) = sumdiv(n, d, isb(d)); \\ Michel Marcus, Jan 24 2021

A341057 Numbers without Brazilian divisors.

Original entry on oeis.org

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

Bernard Schott, Feb 04 2021

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).

			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.

Cf. A125134, A340795, A308851, A341058 (with 1 Brazilian divisor).
Subsequence of A220570 (non-Brazilian numbers).
Supersequence of A220627 (non-Brazilian primes).

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 *)

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

A340795(a(n)) = 0.

cp's OEIS Frontend

A340795 a(n) is the number of divisors of n that are Brazilian.

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