A340796 -id:A340796 - 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

A340797 Integers whose number of divisors that are Brazilian sets a new record.

Original entry on oeis.org

1, 7, 14, 24, 40, 48, 60, 84, 120, 168, 240, 336, 360, 420, 672, 720, 840, 1260, 1680, 2520, 3360, 5040, 7560, 10080, 15120, 20160, 25200, 27720, 30240, 43680, 45360, 50400, 55440, 65520, 83160, 98280, 110880, 131040, 166320, 196560, 221760, 262080, 277200, 327600

Offset: 1

Bernard Schott, Jan 24 2021

Corresponding number of Brazilian divisors: 0, 1, 2, 3, 4, 5, 6, 7, 10, 11, 14, 15, 17, 18, 19, 26, ...

Observation: the 58 consecutive highly composite numbers from A002182(12) = 240 to A002182(69) = 2095133040 (maybe more, according to conjectured terms) are also terms of this sequence.

			40 has 8 divisors: {1, 2, 4, 5, 8, 10, 20, 40} of which 4 are Brazilian: {8, 10, 20, 40}. No positive integer smaller than 40 has as many as four Brazilian divisors; hence 40 is a term.

David A. Corneth, Table of n, a(n) for n = 1..85
David A. Corneth, Conjectured terms
Wikipédia, Nombre brésilien (in French).
Index entries for sequences related to Brazilian numbers.

Cf. A002182, A125134, A220570, A308851, A340795, A340796.

Similar with: A093036 (palindromes), A340548 (repdigits), A340549 (repunits), A340637 (Niven), A340638 (Zuckerman).

brazQ[n_] := Module[{b = 2, found = False}, While[b < n - 1 && Length[Union[ IntegerDigits[n, b]]] > 1, b++]; b < n - 1]; d[n_] := DivisorSum[n, 1 &, brazQ[#] &]; dm = -1; s = {}; Do[d1 = d[n]; If[d1 > dm, dm = d1; AppendTo[s, n]], {n, 1, 1000}]; s (* Amiram Eldar, Jan 24 2021 *)

isb(n) = for(b=2, n-2, d=digits(n, b); if(vecmin(d)==vecmax(d), return(1))); \\ A125134
nbd(n) = sumdiv(n, d, isb(d)); \\ A340795
lista(nn) = {my(m=-1); for (n=1, nn, my(x = nbd(n)); if (x > m, print1(n, ", "); m = x););} \\ Michel Marcus, Jan 24 2021

a(20)-a(36) from Michel Marcus, Jan 24 2021

a(37)-a(44) from Amiram Eldar, Jan 24 2021

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

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