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A362842 a(1) = 1; a(2) = 2; for n > 2, a(n) is the smallest positive number that has not yet appeared that shares a factor with a(n-1) when both a(n-1) and a(n) are read as numbers in bases from one more than the maximum digit in a(n-1) and a(n), up to base 10.

%I A362842 #16 May 07 2023 08:29:56
%S A362842 1,2,4,6,3,9,12,24,8,20,10,30,33,11,22,26,13,39,15,48,28,14,49,7,70,
%T A362842 16,38,19,57,69,18,56,76,36,60,40,42,21,63,66,44,46,23,92,32,64,62,31,
%U A362842 93,27,90,5,50,55,77,84,35,80,68,17,119,34,94,47,329,91,52,96,45,95,25,190,54,98,58,29
%N A362842 a(1) = 1; a(2) = 2; for n > 2, a(n) is the smallest positive number that has not yet appeared that shares a factor with a(n-1) when both a(n-1) and a(n) are read as numbers in bases from one more than the maximum digit in a(n-1) and a(n), up to base 10.
%C A362842 This is a base variation of the EKG sequence A064413. Despite numbers with larger digits having to share a factor with a(n-1) in fewer bases than those with only small digits, and would therefore seemingly appear more frequently, the frequency of the digits 8 and 9, for example, in the first 200000 terms is the same as the smaller digits 0 to 7, so surprisingly this does not appear to influence the determination of a(n).
%C A362842 In the first 200000 terms the smallest unused number is 25411, which implies all numbers will eventually appear. In the same range the fixed points are 1, 2, 424, 507, 1261, 1577, 2461, 4311; it is likely no more appear.
%H A362842 Scott R. Shannon, <a href="/A362842/b362842.txt">Table of n, a(n) for n = 1..10000</a>
%H A362842 Scott R. Shannon, <a href="/A362842/a362842.png">Image of the first 100000 terms</a>. The green line is a(n) = n.
%e A362842 a(7) = 12 as the maximum digit in a(6) = 9 and 12 is 9, so a(6) and a(7) are only read as base 10 numbers, and 12 is the smallest unused number which shares a factor with 9 in base 10.
%e A362842 a(8) = 24 as the maximum digit in a(7) = 12 and 24 is 4, and 12_k shares a factor with 24_k when they are read as numbers in all bases k = 5,6,7,8,9,10. No unused smaller number has this property, e.g. a(8) cannot equal 8 as a(7) in base 9 is 12_9 = 11, which does not share a factor with 8_9 = 8. This is the first term to differ from A064413.
%e A362842 a(9) = 8 as the maximum digit in a(8) = 24 and 8 is 8, and 24_k shares a factor with 8_k when they are read as numbers in all bases k = 9,10.
%Y A362842 Cf. A064413, A004053, A354087, A352763, A348086, A337687.
%K A362842 nonn,base
%O A362842 1,2
%A A362842 _Scott R. Shannon_, May 05 2023