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%I #85 Mar 02 2019 04:22:27
%S 1,3,17,41,10,6,77,33,7,8,28,167,1292,382,58,14,37,192,97,89,94,59,26,
%T 161,141,1187,71,22,148,3847,63,79,281,95,308,66,81,90,57,2387,288,
%U 1697,319,1786,669,30,173,1315,3626,924,20,447,67,2588,352,593,418,86,293,98
%N The decimal expansion of the fractional part of a(n)/a(n+1) starts with a(n+1) (disregarding leading zeros); always choose the smallest possible positive integer not occurring earlier.
%C Numbers which can only appear as the first term of this sequence or the corresponding variant: 1, 2, 4, 5, 9, 11, 12, 13, 15, 16, 18, 19, 21, 23, 24, 25, 27, 29, 31, 32, 34, 35, 38, 39, 43, 44, 45, 46, 47, 48, 49, etc., i.e., A298981. - _Robert G. Wilson v_, Jan 17 2018
%C The sequence is infinite. There will always be a solution of the form floor(sqrt(a(n)*10^k)) with k sufficiently large (namely, choose k such that this is larger than a(n) and the fractional part is < 0.5). - _M. F. Hasler_, Jan 17 2018
%C a(2456) > 600000000. - _Robert G. Wilson v_, Jan 18 2018
%C a(2456) <= 7581556568. - _M. F. Hasler_, Jan 19 2018
%C If the constraint that a(n) be a term not occurring earlier were removed, the sequence would cycle {3, 17, 41, 10}. - _Robert G. Wilson v_, Feb 04 2018
%C Records: 1, 3, 17, 41, 77, 167, 1292, 3847, 80498, 83666, 390256, 536097, 886566, 2533515, 4881598, 275680975, 7581556568, 10669182255, 31559467676, ... - _Robert G. Wilson v_, Feb 05 2018
%H Robert G. Wilson v, <a href="/A298232/b298232.txt">Table of n, a(n) for n = 1..10661</a> (first 1001 terms from Jean-Marc Falcoz)
%e 1 divided by 3 is 0.3333333333... which shows "3" immediately after the decimal point;
%e 3 divided by 17 is 0.1764705882... which shows "17" immediately after the decimal point;
%e 17 divided by 41 is 0.4146341463... which shows "41" immediately after the decimal point;
%e 41 divided by 10 is 4.1000000000... which shows "10" immediately after the decimal point;
%e 10 divided by 6 is 1.6666666666... which shows "6" immediately after the decimal point;
%e 6 divided by 77 is 0.07792207792... which shows "77" after the decimal point and the leading zero;
%e etc.
%t f[s_List] := Block[{k = 2, m = s[[-1]]}, While[k = g[k, m]; MemberQ[s, k], k++]; Append[s, k]]; g[k_, m_] := Block[{j, l = k}, While[j = 10^IntegerLength[l]*Mod[m, l]/l; While[0 < Floor@j < l, j *= 10]; Floor[j] != l, l++]; l]; Nest[f, {1}, 100] (* _Robert G. Wilson v_, Jan 16 2018 and revised Jan 31 2018 *)
%o (PARI) {u=[a=1]; (nxt()=for(b=u[1]+1,oo, !setsearch(u,b) && (f=frac(a/b)) && f\10^(-logint((b-1)\f,10)-1)==b&&return(b))); for(i=2,200, print1(a,","); u=setunion(u,[a=nxt()]));a} \\ _M. F. Hasler_, Jan 17 2018
%Y Cf. A298980, A298981, A298982.
%K nonn,base
%O 1,2
%A _Eric Angelini_ and _Jean-Marc Falcoz_, Jan 15 2018
%E Corrected by _Rémy Sigrist_ and _Jacques Tramu_, Jan 16 2018
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