A270034 -id:A270034 - cp's OEIS frontend

A216194 a(n) = Smallest b for which the base b representation of n contains at least one 2 (or 0 if no such base exists).

0, 3, 0, 0, 3, 3, 3, 3, 4, 4, 3, 5, 5, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 10, 3, 4, 11, 3, 3, 3, 3, 4, 4, 3, 4, 4, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 6, 4, 3, 6, 5, 3, 3, 3, 3, 4, 4, 3, 6, 4, 3, 3, 3, 3, 3, 3

Offset: 1

Nathan Fox, Mar 12 2013

a(n)=3 if and only if n is in A074940.
a(n) > 0 for n >= 5 since 12 is n written in base n-2.
The only perfect k-th powers (k>=2) that can appear in this sequence are 2^k with k a prime number.
The first n for which a(n)=7 is 849.
The first n for which a(n)=8 is 1084.
The first n for which a(n)=10 is 28.  The second is 243.
The first n for which a(n)=11 is 31.  The second is 58130496.
a(n)<=11 for all n with fewer than 3000 base 10 digits.  No n for which a(n)>11 has been found.

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

Cf. A216192, A270027, A270028, A270029, A270030, A270031, A270032, A270033, A270034, A270035.

firstNTerms:=proc(n) local b,i,rep,L:
L:=[]:
for i from 5 to n do
    b:=3:
    while true do
        rep:=convert(i, base, b):
        if evalb(2 in rep) then
            L:=[op(L), b]:
            break:
        fi:
        b:=b+1:
    od:
od:
L:
end:

sb2[n_]:=Module[{b=3},While[DigitCount[n,b,2]<1,b++];b]; Array[sb2,110,5] (* Harvey P. Dale, Jan 16 2016 *)
Table[SelectFirst[Range[3, 1200], DigitCount[n, #, 2] > 0 &], {n, 5, 120}] (* Michael De Vlieger, Mar 09 2016, Version 10 *)

a(n) = if ((n<5) && (n!=2), 0, my(b=3); while (! vecsearch(vecsort(digits(n, b)), 2), b++); b);  \\ Michel Marcus, Aug 06 2014, Mar 11 2016

Modified the definition to make the offset 1 by Nathan Fox, Mar 10 2016

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).

Original entry on oeis.org

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

A270028 a(n) is the smallest b >= 3 for which the base-b representation of n contains at least one 1 (or 0 if no such base exists).

Original entry on oeis.org

3, 0, 3, 3, 3, 4, 3, 5, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 3, 4, 3, 3, 3, 4, 3, 4, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 3, 5, 3, 3, 3, 6, 3, 6, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 3, 4, 3, 3, 3, 4, 3, 4, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3

Offset: 1

Nathan Fox, Mar 08 2016

If we drop the b >= 3 requirement, then this sequence becomes A007395 (the constant 2 sequence).
a(n) > 0 for n >= 3 since the base-(n-1) representation of n is 11.
a(n)=3 if and only if n is in A081606.
The only perfect k-th powers (k >= 2) that can appear in this sequence are 2^k with k a prime number.
The first n for which a(n)=7 is 560.
The first n for which a(n)=8 is 870899850.
The first n for which a(n)=10 is 871017138.
The first n for which a(n)=11 is 65473886952.
The first n for which a(n)=12 is 65473886954.

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

Cf. A270027, A216194, A270029, A270030, A270031, A270033, A270034, A270035, A270038.

Table[SelectFirst[Range[3, 10], DigitCount[n, #, 1] > 0 &], {n, 3, 120}] (* Michael De Vlieger, Mar 10 2016, Version 10 *)

a(n) = if (n==2, 0, my(b=3); while(!vecsearch(Set(digits(n, b)), 1), b++); b); \\ Michel Marcus, Mar 10 2016

A270029 a(n) is the smallest b for which the base-b representation of n contains at least one 3 (or 0 if no such base exists).

Original entry on oeis.org

0, 0, 4, 0, 0, 0, 4, 5, 6, 7, 4, 4, 4, 4, 4, 5, 5, 5, 4, 6, 6, 6, 4, 7, 7, 7, 4, 4, 4, 4, 4, 9, 5, 9, 4, 10, 10, 5, 4, 5, 5, 5, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 17, 5, 5, 4, 5, 5, 7, 4, 7, 5, 7, 4, 4, 4, 4, 4, 5, 5, 5, 4, 5, 5, 5, 4, 5, 5, 5, 4, 4, 4, 4, 4, 5, 5, 5, 4, 26

Offset: 1

Nathan Fox, Mar 08 2016

a(n) > 0 for n >= 7 since 13 is n written in base n-3.

The only perfect k-th powers (k >= 2) that can appear in this sequence are 2^k and 3^k with k a prime number.

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

Cf. A270027, A270028, A216194, A270030, A270031, A270032, A270033, A270034, A270035, A270039.

Table[SelectFirst[Range[4, 10^3], DigitCount[n, #, 3] > 0 &], {n, 7, 120}] (* Michael De Vlieger, Mar 10 2016, Version 10 *)

a(n) = if ((n<7) && (n!=3), 0, my(b=4); while(!vecsearch(Set(digits(n, b)), 3), b++); b); \\ Michel Marcus, Mar 10 2016

A270030 a(n) is the smallest b for which the base-b representation of n contains at least one 4 (or 0 if no such base exists).

Original entry on oeis.org

0, 0, 0, 5, 0, 0, 0, 0, 5, 6, 7, 8, 9, 5, 11, 6, 13, 7, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 5, 7, 7, 7, 7, 5, 8, 8, 8, 8, 5, 6, 9, 9, 9, 5, 5, 5, 5, 5, 5, 11, 11, 6, 7, 5, 12, 12, 12, 6, 5, 6, 6, 6, 6, 5, 6, 14, 7, 8, 5, 5, 5, 5, 5, 5, 16, 6, 7, 7, 5, 7, 7, 6, 7, 5, 9, 18, 18, 6, 5, 19

Offset: 1

Nathan Fox, Mar 08 2016

a(n) > 0 for n >= 9 since 14 is n written in base n-4.

The only perfect k-th powers (k >= 2) that can appear in this sequence are 2^k, 3^k, or 4^k, with k a prime number.

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

Cf. A268236, A268237, A270027, A270028, A216194, A270029, A270031, A270032, A270033, A270034, A270035, A270040.

Table[SelectFirst[Range[5, 10^3], DigitCount[n, #, 4] > 0 &], {n, 9, 120}] (* Michael De Vlieger, Mar 10 2016, Version 10 *)

a(n) = if ((n<9) && (n!=4), 0, my(b=5); while(!vecsearch(Set(digits(n, b)), 4), b++); b); \\ Michel Marcus, Mar 10 2016

A270031 a(n) is the smallest b for which the base-b representation of n contains at least one 5 (or 0 if no such base exists).

Original entry on oeis.org

0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 6, 7, 8, 9, 10, 11, 6, 13, 7, 15, 8, 17, 6, 19, 10, 7, 11, 23, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 6, 8, 8, 8, 8, 8, 6, 9, 9, 9, 9, 9, 6, 7, 10, 10, 10, 10, 6, 11, 7, 11, 11, 11, 6, 6, 6, 6, 6, 6, 6, 13, 13, 13, 7, 13, 6, 14, 14, 14, 14, 7, 6, 7, 7, 7, 7, 7

Offset: 1

Nathan Fox, Mar 08 2016

a(n) > 0 for n >= 11 since 15 is n written in base n-5.

The only perfect k-th powers (k >= 2) that can appear in this sequence are m^k with 2 <= m <= 5 and k a prime number.

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

Cf. A270027, A270028, A216194, A270029, A270030, A270032, A270033, A270034, A270035, A270041.

Table[SelectFirst[Range[6, 10^3], DigitCount[n, #, 5] > 0 &], {n, 11, 120}] (* Michael De Vlieger, Mar 10 2016, Version 10 *)

a(n) = if ((n<11) && (n!=5), 0, my(b=6); while(!vecsearch(Set(digits(n, b)), 5), b++); b); \\ Michel Marcus, Mar 10 2016

A270033 a(n) is the smallest b for which the base-b representation of n contains at least one 7 (or 0 if no such base exists).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 8, 9, 10, 11, 12, 13, 14, 15, 8, 17, 9, 19, 10, 21, 11, 23, 8, 25, 13, 9, 14, 29, 10, 31, 8, 11, 17, 35, 9, 37, 19, 13, 8, 41, 14, 43, 11, 9, 23, 47, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 8, 10, 10, 10, 10, 10, 10, 10, 8, 11, 11

Offset: 1

Nathan Fox, Mar 08 2016

a(n) > 0 for n >= 15 since 17 is n written in base n-7.

The only perfect k-th powers (k >= 2) that can appear in this sequence are m^k with 2 <= m <= 7 and k a prime number.

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

Cf. A270027, A270028, A216194, A270029, A270030, A270031, A270032, A270034, A270035, A270043.

Table[SelectFirst[Range[8, 1200], DigitCount[n, #, 7] > 0 &], {n, 15, 120}] (* Michael De Vlieger, Mar 10 2016, Version 10 *)

a(n) = if ((n<15) && (n!=7), 0, my(b=8); while(!vecsearch(Set(digits(n, b)), 7), b++); b); \\ Michel Marcus, Mar 10 2016

A270035 a(n) is the smallest b for which the base-b representation of n contains at least one 9 (or 0 if no such base exists).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 10, 21, 11, 23, 12, 25, 13, 27, 14, 29, 10, 31, 16, 11, 17, 35, 12, 37, 19, 13, 10, 41, 14, 43, 11, 15, 23, 47, 12, 49, 10, 17, 13, 53, 18, 11, 14, 19, 29, 59, 10, 61, 31, 21, 16, 13, 11, 67, 17, 23, 10, 71

Offset: 1

Nathan Fox, Mar 08 2016

a(n) > 0 for n >= 19 since 19 is n written in base n-9.

The only perfect k-th powers (k >= 2) that can appear in this sequence are m^k with 2 <= m <= 9 and k a prime number.

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

Cf. A270027, A270028, A216194, A270029, A270030, A270031, A270032, A270033, A270034, A270045.

Table[SelectFirst[Range[10, 1200], DigitCount[n, #, 9] > 0 &], {n, 19, 120}] (* Michael De Vlieger, Mar 10 2016, Version 10 *)

a(n) = if ((n<19) && (n!=9), 0, my(b=10); while(!vecsearch(Set(digits(n, b)), 9), b++); b); \\ Michel Marcus, Mar 10 2016

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 *)

A270032 a(n) is the smallest b for which the base-b representation of n contains at least one 6 (or 0 if no such base exists).

Original entry on oeis.org

0, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, 7, 8, 9, 10, 11, 12, 13, 7, 15, 8, 17, 9, 19, 10, 7, 11, 23, 8, 25, 13, 9, 7, 29, 10, 31, 8, 11, 17, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 7, 9, 9, 9, 9, 9, 9, 7, 10, 10, 10, 10, 10, 10, 7, 8, 11, 11, 11, 11, 11, 7, 12, 8, 12, 12, 12

Offset: 1

Nathan Fox, Mar 08 2016

a(n) > 0 for n >= 13 since 16 is n written in base n-6.

The only perfect k-th powers (k >= 2) that can appear in this sequence are m^k with 2 <= m <= 6 and k a prime number.

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

Cf. A270027, A270028, A216194, A270029, A270030, A270031, A270033, A270034, A270035, A270042.

Table[SelectFirst[Range[7, 1200], DigitCount[n, #, 6] > 0 &], {n, 13, 120}] (* Michael De Vlieger, Mar 10 2016, Version 10 *)

a(n) = if ((n<13) && (n!=6), 0, my(b=7); while(!vecsearch(Set(digits(n, b)), 6), b++); b); \\ Michel Marcus, Mar 10 2016

cp's OEIS Frontend

A216194 a(n) = Smallest b for which the base b representation of n contains at least one 2 (or 0 if no such base exists).

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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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A270028 a(n) is the smallest b >= 3 for which the base-b representation of n contains at least one 1 (or 0 if no such base exists).

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A270029 a(n) is the smallest b for which the base-b representation of n contains at least one 3 (or 0 if no such base exists).

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A270030 a(n) is the smallest b for which the base-b representation of n contains at least one 4 (or 0 if no such base exists).

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A270031 a(n) is the smallest b for which the base-b representation of n contains at least one 5 (or 0 if no such base exists).

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A270033 a(n) is the smallest b for which the base-b representation of n contains at least one 7 (or 0 if no such base exists).

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A270035 a(n) is the smallest b for which the base-b representation of n contains at least one 9 (or 0 if no such base exists).

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