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A298232
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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.
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4
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1, 3, 17, 41, 10, 6, 77, 33, 7, 8, 28, 167, 1292, 382, 58, 14, 37, 192, 97, 89, 94, 59, 26, 161, 141, 1187, 71, 22, 148, 3847, 63, 79, 281, 95, 308, 66, 81, 90, 57, 2387, 288, 1697, 319, 1786, 669, 30, 173, 1315, 3626, 924, 20, 447, 67, 2588, 352, 593, 418, 86, 293, 98
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OFFSET
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1,2
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COMMENTS
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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
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
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
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
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LINKS
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EXAMPLE
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1 divided by 3 is 0.3333333333... which shows "3" immediately after the decimal point;
3 divided by 17 is 0.1764705882... which shows "17" immediately after the decimal point;
17 divided by 41 is 0.4146341463... which shows "41" immediately after the decimal point;
41 divided by 10 is 4.1000000000... which shows "10" immediately after the decimal point;
10 divided by 6 is 1.6666666666... which shows "6" immediately after the decimal point;
6 divided by 77 is 0.07792207792... which shows "77" after the decimal point and the leading zero;
etc.
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,base
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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