A069575 -id:A069575 - 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

A270037 a(n) = Smallest m >= 3 containing no zeros when represented in any base from 3 through n.

Original entry on oeis.org

4, 5, 7, 7, 13, 13, 13, 13, 13, 13, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 43, 43, 43, 43, 43, 43, 43, 43, 43, 43, 43, 43, 43, 43, 43, 43, 43, 43, 43, 43, 157, 157, 157, 157, 157, 157, 157, 157, 157, 157, 157, 157, 157, 157, 157, 157, 157, 157, 157, 157, 157, 157, 157

Offset: 3

Nathan Fox, Mar 09 2016

It remains to be determined if the sequence is finite.

Every known term in this sequence is in A069575, and every term in A069575 is in this sequence.

Nathan Fox, Table of n, a(n) for n = 3..618

Cf. A069575, A270027, A270038, A216192, A270039, A270040, A270041, A270042, A270043, A270044, A270045.

Table[SelectFirst[Range[3, 10^3], Total@ Map[Function[k, DigitCount[#, k, 0]], Range[3, n]] == 0 &], {n, 3, 80}] /. n_ /; MissingQ@ n -> Nothing (* Michael De Vlieger, Mar 10 2016, Version 10.2 *)

isok(m, n) = {for (b=3, n, if (! vecmin(digits(m, b)), return (0));); 1;}
a(n) = {my(m = 3); while (! isok(m,n), m++); m;} \\ Michel Marcus, Mar 10 2016

A277779 Numbers k that only contain a 0 in bases that divide k.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 12, 14, 15, 16, 22, 24, 30, 36, 42, 44, 48, 60, 63, 72, 80, 90, 96, 120, 156, 158, 160, 186, 192, 212, 214, 216, 222, 240, 288, 312, 420, 432, 468, 474, 480, 484, 618, 620, 624, 840, 942, 948, 960, 996, 1124, 1200, 1224, 1494, 1560, 1656

Offset: 1

Bobby Jacobs, Oct 30 2016

Probably finite.

			6 only contains a 0 in bases 2, 3, and 6. Those bases divide 6. Therefore, 6 is in this sequence.

Chai Wah Wu, Table of n, a(n) for n = 1..261 (all terms < 10^11; first 250 terms from Charles R Greathouse IV)

Cf. A069575.

is(n)=for(b=2,sqrtint(n-1), if(Set(digits(n,b))[1]==0 && n%b, return(0))); 1 \\ Charles R Greathouse IV, Oct 31 2016

More terms from Alois P. Heinz, Oct 30 2016

A319033 a(n) is the (conjectured) largest number k that is zeroless in every base b such that n <= b < k.

Original entry on oeis.org

7, 619, 26237, 698531, 3979433, 3979433, 29643151199, 29643151199, 29643151199, 29643151199, 260621258159, 260621258263, 260621258263, 296126238241, 296126238241, 296126238241, 296126238241, 556715917481, 971156053631, 971156053631, 971156053631, 971156053631

Offset: 2

Jon E. Schoenfield, Oct 08 2018

All terms are necessarily prime.
It seems nearly certain that there is no k > 7 that is zeroless in every base from 2 through k-1; if such a k exists, it exceeds 2^(10^9).
Up to 10^5000 (see A069575), no number k > 619 is zeroless in every base from 3 through k-1.
a(4) = 26237 or > 10^1000; a(5) = 698531 or > 10^1000; a(6) = a(7) = 3979433 unless a(7) > 10^1000; a(8) = a(9) = a(10) = a(11) = 29643151199 unless a(11) > 10^1000; it seems extremely unlikely that any of these terms could actually exceed 10^1000.

			a(2) = 7 because k = 7 = 111_2 = 21_3 = 13_4 = 12_5 = 11_6, with no zero digits in any base from 2 through k-1, and this is almost certainly (see Comments) the largest such number having this property.
a(3) = 619 because k = 619 = 211221_3 = 21223_4 = 4434_5 = 2511_6 = 1543_7 = 1153_8 = 757_9 = 619_10 = 513_11 = 437_12 = 388_13 = 323_14 = 2B4_15 = ... = 11_(k-1), and this is almost certainly (see Comments) the largest number having this property.

Cf. A052382, A069575, A270027, A270037, A277779.

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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A270037 a(n) = Smallest m >= 3 containing no zeros when represented in any base from 3 through n.

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A277779 Numbers k that only contain a 0 in bases that divide k.

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A319033 a(n) is the (conjectured) largest number k that is zeroless in every base b such that n <= b < k.

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