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a(n) is the (conjectured) largest number k that is zeroless in every base b such that n <= b < k.
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%I #24 Oct 23 2018 08:53:31

%S 7,619,26237,698531,3979433,3979433,29643151199,29643151199,

%T 29643151199,29643151199,260621258159,260621258263,260621258263,

%U 296126238241,296126238241,296126238241,296126238241,556715917481,971156053631,971156053631,971156053631,971156053631

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

%C All terms are necessarily prime.

%C 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).

%C Up to 10^5000 (see A069575), no number k > 619 is zeroless in every base from 3 through k-1.

%C 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.

%e 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.

%e 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.

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

%K nonn,base

%O 2,1

%A _Jon E. Schoenfield_, Oct 08 2018