A298232 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.

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

Offset: 1

Author

Eric Angelini and Jean-Marc Falcoz, Jan 15 2018

Keywords

• nonn
• base

Comments

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

a(2456) > 600000000. - Robert G. Wilson v, Jan 18 2018

a(2456) <= 7581556568. - M. F. Hasler, Jan 19 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

Examples

			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.
		

Links

• Robert G. Wilson v, Table of n, a(n) for n = 1..10661 (first 1001 terms from Jean-Marc Falcoz)

Crossrefs

Cf. A298980, A298981, A298982.

Programs

Mathematica

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 *)

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

Extensions

Corrected by Rémy Sigrist and Jacques Tramu, Jan 16 2018