A270037 -id:A270037 - cp's OEIS frontend

A270027 a(n) is the smallest b >= 3 for which the base-b representation of n contains at least one 0 (or 0 if no such base exists).

0, 0, 3, 4, 5, 3, 7, 4, 3, 3, 3, 3, 13, 7, 3, 4, 4, 3, 3, 3, 3, 11, 23, 3, 5, 5, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 6, 3, 43, 4, 3, 3, 3, 3, 4, 4, 3, 4, 5, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 3, 4, 4, 3, 3, 3, 3, 4, 4, 3, 4, 4, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3

Offset: 1

Nathan Fox, Mar 08 2016

It is natural to consider this sequence starting from base 3 instead of base 2, as the latter causes most terms to be 2 (see A270026).
a(n) = n if and only if n = 3 or n is in A069575.
a(n) > 0 for n >= 3 since the base-n representation of n is 10.

Nathan Fox, Table of n, a(n) for n = 1..10000

Cf. A069575, A270026, A270028, A216194, A270029, A270030, A270031, A270033, A270034, A270035, A270037.

Table[SelectFirst[Range[3, 10^3], DigitCount[n, #, 0] > 0 &], {n, 3, 120}] (* Michael De Vlieger, Mar 10 2016, Version 10 *)

a(n) = if (n < 3, 0, my(b=3); while(vecmin(digits(n, b)), b++); b); \\ Michel Marcus, Mar 10 2016

A216192 a(n) = Smallest m >= 5 containing no twos when represented in any base from 3 through n.

Original entry on oeis.org

9, 12, 28, 28, 28, 28, 28, 31

Offset: 3

Nathan Fox, Mar 12 2013

If a(11) exists, it contains over 3000 digits.

No more terms < 10^154400. Most likely there are no more terms. - Chai Wah Wu, Mar 17 2016

			In base 3, 5=12, 6=20, 7=21, 8=22, 9=100.  The first representation containing no twos is that of 9, so a(3)=9.

Nathan Fox, Maple Program to compute a(n)

Cf. A216194, A270037, A270038, A270039, A270040, A270041, A270042, A270043, A270044, A270045.

Table[SelectFirst[Range[5, 10^3], Total@ Map[Function[k, DigitCount[#, k, 2]], Range[3, n]] == 0 &], {n, 3, 20}] /. n_ /; MissingQ@ n -> Nothing (* Michael De Vlieger, Mar 09 2016, Version 10.2 *)

A270039 a(n) = Smallest m >= 7 containing no threes when represented in any base from 4 through n.

Original entry on oeis.org

8, 9, 10, 32, 32, 36, 64, 64, 64, 64, 64, 64, 64, 100, 100, 100, 100, 100, 100, 100, 100, 100, 104, 145, 145, 145, 145, 145, 145, 145, 145, 145, 145

Offset: 4

Nathan Fox, Mar 09 2016

It remains to be determined if the sequence is finite.

Cf. A270029, A216192, A270037 - A270045.

A270039 := proc(n)
    local m,nothree,b;
    for m from 7 do
        nothree := true;
        for b from 4 to n do
            convert(convert(m,base,b),set) ;
            if 3 in % then
                nothree := false;
                break;
            fi;
        end do:
        if nothree then
            return m;
        end if;
    end do:
end proc: # R. J. Mathar, Mar 11 2016

Table[SelectFirst[Range[7, 10^3], Total@ Map[Function[k, DigitCount[#, k, 3]], Range[4, n]] == 0 &], {n, 7, 60}] /. n_ /; MissingQ@ n -> Nothing (* Michael De Vlieger, Mar 10 2016, Version 10.2 *)

A270045 Smallest m >= 19 containing no nines when represented in any base from 10 through n.

Original entry on oeis.org

20, 21, 22, 23, 24, 25, 26, 27, 28, 30, 30, 32, 32, 34, 34, 36, 36, 38, 38, 40, 40, 44, 44, 44, 44, 46, 46, 50, 50, 50, 50, 52, 52, 56, 56, 56, 56, 58, 58, 62, 62, 62, 62, 68, 68, 68, 68, 68, 68, 70, 70, 76, 76, 76, 76, 76, 76, 80, 80, 80, 80, 82, 82, 88, 88, 88, 88, 88, 88, 840

Offset: 10

Nathan Fox, Mar 09 2016

It remains to be determined if the sequence is finite.

Nathan Fox, Table of n, a(n) for n = 10..359

Cf. A270035, A270037, A270038, A216192, A270039, A270040, A270041, A270042, A270043, A270044.

Table[SelectFirst[Range[19, 10^3], Total@ Map[Function[k, DigitCount[#, k, 9]], Range[10, n]] == 0 &], {n, 10, 79}] /. n_ /; MissingQ@ n -> Nothing (* Michael De Vlieger, Mar 10 2016, Version 10.2 *)

A270040 a(n) = Smallest m >= 9 containing no fours when represented in any base from 5 through n.

Original entry on oeis.org

10, 11, 12, 13, 15, 15, 17, 17, 66, 75, 75, 86, 86, 90, 138, 138, 138, 138, 138, 138, 138, 138, 138, 182, 182, 182, 182, 182, 182, 182, 182, 182, 185, 781817578165, 781817578165, 7826560751018861596150680

Offset: 5

Nathan Fox, Mar 09 2016

It remains to be determined if the sequence is finite.

These numbers are not very Foury, at least not initially. (See A268236.)

Nathan Fox, Table of n, a(n) for n = 5..59

Cf. A268236, A268237, A270030, A270037, A270038, A216192, A270039, A270041, A270042, A270043, A270044, A270045.

Table[SelectFirst[Range[9, 10^3], Total@ Map[Function[k, DigitCount[#, k, 4]], Range[5, n]] == 0 &], {n, 5, 60}] /. n_ /; MissingQ@ n -> Nothing (* Michael De Vlieger, Mar 10 2016, Version 10.2 *)

A270041 a(n) = Smallest m >= 11 containing no fives when represented in any base from 6 through n.

Original entry on oeis.org

12, 13, 14, 15, 16, 18, 18, 20, 20, 22, 22, 24, 24, 28, 28, 28, 28, 144, 144, 160, 160, 162, 216, 216, 216, 216, 216, 216, 216, 216, 216, 223, 228, 316, 316, 316, 316, 316, 316, 316, 316, 316, 316, 316, 316, 316, 316, 487, 487

Offset: 6

Nathan Fox, Mar 09 2016

It remains to be determined if the sequence is finite.

Nathan Fox, Table of n, a(n) for n = 6..109

Cf. A270031, A270037, A270038, A216192, A270039, A270040, A270042, A270043, A270044, A270045.

Table[SelectFirst[Range[11, 10^3], Total@ Map[Function[k, DigitCount[#, k, 5]], Range[6, n]] == 0 &], {n, 6, 60}] /. n_ /; MissingQ@ n -> Nothing (* Michael De Vlieger, Mar 10 2016, Version 10.2 *)

A270042 a(n) = Smallest m >= 13 containing no sixes when represented in any base from 7 through n.

Original entry on oeis.org

14, 15, 16, 17, 18, 19, 21, 21, 23, 23, 25, 25, 29, 29, 29, 29, 31, 31, 35, 35, 35, 35, 37, 37, 228, 228, 232, 271, 271, 271, 271, 271, 273, 343, 343, 343, 343, 343, 343, 343, 343, 343, 343, 451, 451, 451, 451, 451, 451, 451, 451, 451, 451, 451, 451, 451, 451, 451, 455, 472, 472, 599

Offset: 7

Nathan Fox, Mar 09 2016

It remains to be determined if the sequence is finite.

Nathan Fox, Table of n, a(n) for n = 7..159

Cf. A270032, A270037, A270038, A216192, A270039, A270040, A270041, A270043, A270044, A270045.

Table[SelectFirst[Range[13, 10^3], Total@ Map[Function[k, DigitCount[#, k, 6]], Range[7, n]] == 0 &], {n, 7, 68}] /. n_ /; MissingQ@ n -> Nothing (* Michael De Vlieger, Mar 10 2016, Version 10.2 *)

A270043 a(n) = Smallest m >= 15 containing no sevens when represented in any base from 8 through n.

Original entry on oeis.org

16, 17, 18, 19, 20, 21, 22, 24, 24, 26, 26, 28, 28, 30, 30, 32, 32, 36, 36, 36, 36, 38, 38, 42, 42, 42, 42, 44, 44, 48, 48, 48, 48, 50, 50, 54, 54, 54, 54, 386, 386, 396, 400, 426, 426, 426, 515, 515, 515, 515, 515, 515, 515, 515, 515, 515, 515, 528

Offset: 8

Nathan Fox, Mar 09 2016

It remains to be determined if the sequence is finite.

Nathan Fox, Table of n, a(n) for n = 8..227

Cf. A270033, A270037, A270038, A216192, A270039, A270040, A270041, A270042, A270044, A270045.

Table[SelectFirst[Range[15, 10^3], Total@ Map[Function[k, DigitCount[#, k, 7]], Range[8, n]] == 0 &], {n, 8, 65}] /. n_ /; MissingQ@ n -> Nothing (* Michael De Vlieger, Mar 10 2016, Version 10.2 *)

A270044 a(n) = Smallest m >= 17 containing no eights when represented in any base from 9 through n.

Original entry on oeis.org

18, 19, 20, 21, 22, 23, 24, 25, 27, 27, 29, 29, 31, 31, 33, 33, 37, 37, 37, 37, 39, 39, 43, 43, 43, 43, 45, 45, 49, 49, 49, 49, 51, 51, 55, 55, 55, 55, 57, 57, 61, 61, 61, 61, 67, 67, 67, 67, 67, 67, 69, 69, 549, 590, 590, 590, 590, 594, 604, 630

Offset: 9

Nathan Fox, Mar 09 2016

It remains to be determined if the sequence is finite.

Nathan Fox, Table of n, a(n) for n = 9..308

Cf. A270034, A270037, A270038, A216192, A270039, A270040, A270041, A270042, A270043, A270045.

Table[SelectFirst[Range[17, 10^3], Total@ Map[Function[k, DigitCount[#, k, 8]], Range[9, n]] == 0 &], {n, 9, 68}] /. n_ /; MissingQ@ n -> Nothing (* Michael De Vlieger, Mar 10 2016, Version 10.2 *)

A270038 a(n) = Smallest m >= 3 containing no ones when represented in any base from 3 through n.

Original entry on oeis.org

6, 8, 60, 560, 870899850, 871017138, 871017138, 65473886952, 65473886954

Offset: 3

Nathan Fox, Mar 09 2016

It remains to be determined if the sequence is finite.

a(12) > 3.4*10^30, if it exists. - Giovanni Resta, Oct 14 2019

Cf. A270028, A270037, A216192, A270039, A270040, A270041, A270042, A270043, A270044, A270045.

Table[SelectFirst[Range[3, 10^3], Total@ Map[Function[k, DigitCount[#, k, 1]], Range[3, n]] == 0 &], {n, 3, 7}] /. n_ /; MissingQ@ n -> Nothing (* Michael De Vlieger, Mar 10 2016, Version 10.2 *)

cp's OEIS Frontend

A270027 a(n) is the smallest b >= 3 for which the base-b representation of n contains at least one 0 (or 0 if no such base exists).

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A216192 a(n) = Smallest m >= 5 containing no twos when represented in any base from 3 through n.

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A270039 a(n) = Smallest m >= 7 containing no threes when represented in any base from 4 through n.

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A270045 Smallest m >= 19 containing no nines when represented in any base from 10 through n.

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A270040 a(n) = Smallest m >= 9 containing no fours when represented in any base from 5 through n.

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A270041 a(n) = Smallest m >= 11 containing no fives when represented in any base from 6 through n.

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A270042 a(n) = Smallest m >= 13 containing no sixes when represented in any base from 7 through n.

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A270043 a(n) = Smallest m >= 15 containing no sevens when represented in any base from 8 through n.

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A270044 a(n) = Smallest m >= 17 containing no eights when represented in any base from 9 through n.

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