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

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

Author

Bernard Schott, Jan 24 2021

Keywords

• nonn
• base

Comments

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.

Examples

			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.
		

Links

• 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.

Crossrefs

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

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

Programs

Mathematica

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 *)

PARI

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

Extensions

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

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