A066460 -id:A066460 - cp's OEIS frontend

A220136 Number of ways that a number n can be written as ddd...d where d is a digit in base b with 1 < b < n-1.

0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 1, 1, 2, 1, 0, 2, 0, 1, 2, 1, 0, 3, 0, 2, 1, 2, 0, 2, 2, 2, 1, 1, 1, 3, 0, 1, 1, 4, 0, 3, 1, 2, 2, 1, 0, 4, 0, 2, 1, 2, 0, 3, 1, 2, 2, 1, 0, 5, 0, 2, 4, 2, 1, 3, 0, 2, 1, 3, 0, 4, 1, 1, 2, 2, 1, 3, 0, 5, 1, 1, 0, 5, 2, 2, 1

Offset: 1

T. D. Noe, Dec 26 2012

nonn

When a(n) > 0, n is called Brazilian (A125134). The first number having exactly k representations is A284758(k) for k >= 0 or A066460(k+1) for k > 0. - Bernard Schott, Apr 08 2017

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A066460, A125134, A284758.

brazBases[n_] := Select[Range[2, n - 2], Length[Union[IntegerDigits[n, #]]] == 1 &]; Table[Length[brazBases[n]], {n, 100}]

a(n) = sum(i=2, n-2, #vecsort(digits(n,i), , 8)==1) \\ David A. Corneth, Apr 08 2017

A284758 The least positive integer that has exactly n different representations as Brazilian number.

Original entry on oeis.org

1, 7, 15, 24, 40, 60, 144, 120, 180, 336, 420, 360, 900, 960, 720, 840, 1260, 1440, 2340, 1680, 2880, 3600, 8190, 2520, 9072, 9900, 6300, 6720, 20592, 5040, 10920, 7560, 31320, 98040, 25920, 10080, 21420, 177156, 74256, 15120, 28560, 20160

Offset: 0

Bernard Schott, Apr 04 2017

The representation n = 11_(n-1) is not accepted under the definition of a Brazilian number.

The records of this sequence are the highly Brazilian numbers; hence, this sequence is a supersequence of A329383.

			a(0) = 1 because 1 is the smallest non-Brazilian number.
a(4) = 40 because 40 = 1111_3 = 55_7 = 44_9 = 22_19 and 40 is the smallest integer with four Brazilian representations.

D. Lignon, Dictionnaire de (presque) tous les nombres entiers, Ellipses, 2012, page 420.

Giovanni Resta, Table of n, a(n) for n = 0..100

Cf. A066460, A125134, A329383.

rep[n_] := Length@ Select[Range[2, n/2], 1 == Length@ Union@ IntegerDigits[n, #] &]; a[n_] := Block[{k=1}, While[rep[k] != n, k++]; k]; a /@ Range[0, 15] (* Giovanni Resta, Apr 04 2017 *)

A066044 Numbers k that are repdigits in more bases (smaller than k) than any smaller number.

Original entry on oeis.org

1, 3, 7, 15, 24, 40, 60, 120, 180, 336, 360, 720, 840, 1260, 1440, 1680, 2520, 5040, 7560, 10080, 15120, 20160, 25200, 27720, 45360, 50400, 55440, 83160, 110880, 166320, 221760, 277200, 332640, 498960, 554400, 665280, 720720, 1081080, 1441440, 2162160, 2882880

Offset: 1

Erich Friedman, Dec 29 2001

A repdigit has all digits the same in some base.

The number 3 isn't Brazilian (A125134) because 3 = 11_2 is an expansion of the type n = 11_(n-1), which is forbidden for Brazilian numbers. So, except for 3, all the terms here are highly Brazilian numbers (A329383). - Daniel Lignon, Dec 30 2019

			15 is in the sequence since 15 = 1111_2 = 33_4 = 11_14 and no smaller number is a repdigit in 3 different bases.

D. Lignon, Dictionnaire de (presque) tous les nombres entiers, Editions Ellipses, 2012, see p. 420. [In French.]

Giovanni Resta, Table of n, a(n) for n = 1..92

Cf. A066460, A329383.

a = 0 Range[100]; Do[ c = 1; k = 2; While[ k < n-1, If[ Length[ Union[ IntegerDigits[n, k]]] == 1, c++ ]; k++ ]; If[a[[c]] == 0, a[[c]] = n; Print[c, " = ", n]], {n, 1, 200000} ]

okrepu3(b, target, lim) = {my(k = 3, nb = 0, x); while ((x=(b^k-1)/(b-1)) <= target, if (x==target, nb++); k++); nb;}
dge3(n, d) = {my(nb=0, ndi, limi); for (i=1, #d, ndi = n/d[i]; limi = sqrtint(ndi); for (k=d[i]+1, limi, nb += okrepu3(k, ndi, limi););); nb;}
deq2(n, d) = {my(nb=0, nk); for (k=1, #d\2, nk = (n - d[k])/d[k]; if (nk > d[k], nb++);); nb;}
beta23(n) = {if (n<3, return (0)); my(d=divisors(n)); deq2(n, d) + dge3(n, d);}
lista(nn) = {my(m = -1, nm); for (n=1, nn, if ((nm=beta23(n)) > m, print1(n, ", "); m = nm););} \\ Michel Marcus, Jul 13 2019

More terms from Robert G. Wilson v, Jan 02 2002
Offset changed to 1 by Giovanni Resta, Apr 05 2017
a(1) changed to 1 and new terms a(32)-a(41) from Giovanni Resta, Apr 05 2017

A242397 a(n) is the number of different bases b such that the Brazilian numbers A125134(n) remain a repdigit number.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 3, 2, 1, 2, 2, 2, 2, 1, 1, 1, 3, 1, 1, 4, 3, 1, 2, 2, 1, 4, 2, 1, 2, 3, 1, 2, 2, 1, 5, 2, 4, 2, 1, 3, 2, 1, 3, 4, 1, 1, 2, 2, 1, 3, 5, 1, 1, 5, 2, 2, 1, 3, 4, 2, 2, 2, 1, 1, 5, 2, 2, 3, 3, 3, 3, 1, 5, 2, 2, 4, 4, 1, 2, 2, 1

Offset: 1

Michel Lagneau, May 13 2014

For all numbers m, we restrict the bases b with 1 < b < m-1 because m is repdigit in bases 1 and also m-1.

			a(89) = 7 because A125134(89)=120 and the number 120 is AA in base 11 where A = 10, 88 in base 14, 66 in base 19, 55 in base 23, 44 in base 29, 33 in base 39 and 22 in base 59 => 7 representations.

Michel Lagneau, Table of n, a(n) for n = 1..5000

Cf. A066460, A125134.

for n from 1 to 200 do:c:=0:for b from 2 to n-2 do:x:=convert(n,base,b):n1:=nops(x):a:=x[n1]:i:=1:for k from n1-1 by -1 to 1 do:if x[k]=a then i:=i+1:else fi:od:if i=n1 then c:=c+1:i:=1:else fi:od:if c>0 then printf(`%d, `,c):else fi:od:

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A220136 Number of ways that a number n can be written as ddd...d where d is a digit in base b with 1 < b < n-1.

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A284758 The least positive integer that has exactly n different representations as Brazilian number.

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A066044 Numbers k that are repdigits in more bases (smaller than k) than any smaller number.

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A242397 a(n) is the number of different bases b such that the Brazilian numbers A125134(n) remain a repdigit number.

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