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A096098 a(1) = 2, a(2) = 1; for n >= 3, a(n) = least number not included earlier that divides the concatenation of all previous terms.

%I A096098 #43 Feb 24 2024 01:09:44
%S A096098 2,1,3,71,7,21,599,173,11,23,161,49,13,9,131,19,33,17,1489,331,3989,
%T A096098 69,3097350956401900335673788279883089441874368101,349387,5651,443,29,
%U A096098 51,479470832244949,661,1129,1873,181,1544577973887516219070997863,521
%N A096098 a(1) = 2, a(2) = 1; for n >= 3, a(n) = least number not included earlier that divides the concatenation of all previous terms.
%C A096098 Conjecture (1) Every concatenation is squarefree.
%C A096098 Conjecture (2) This is a rearrangement of the squarefree numbers not divisible by 5. False! (The a(n) are not always squarefree, since a(12)=49 and a(14)=9.)
%C A096098 Fact: All a(n) for n >= 2 are odd, since a(2) = 1 and odd a(n) => odd concatenation => odd a(n+1). - _Wolfdieter Lang_, May 08 2014 (editing an earlier statement).
%C A096098 Conjecture (3) the sequence for n>=2 is a permutation of the positive integers not divisible by 2 or 5.
%C A096098 a(29) is probably 479470832244949, in which case the sequence continues 479470832244949, 661, 1129, 1873, 181. - _Martin Fuller_, Nov 21 2007
%C A096098 Factorization for a(29): 479470832244949*3*17*43217123024009614997922599713504735424547343*P51. - _Sean A. Irvine_, May 25 2010
%C A096098 Assuming Conjecture (3), the smallest number yet to appear is 89. - _Sean A. Irvine_, May 11 2014
%C A096098 The factorization given by Sean A. Irvine above is not for the prime a(29) = 479470832244949 but for the concatenation of a(1), a(2), ..., a(29), and P51 means a prime with 51 digits, namely 202232656574589264871780464738430216507933940172343. - _Wolfdieter Lang_, May 11 2014
%H A096098 Sean A. Irvine, <a href="/A096098/b096098.txt">Table of n, a(n) for n = 1..172</a>
%H A096098 Sean A. Irvine, <a href="/A096098/a096098_2.txt">Factorizations, for n = 1..182</a>
%e A096098 a(6) = 21 as 213717 = 3*7*10177, and 3 = a(3) and 7 = a(4), hence 3*7 = 21 is the least number dividing 213717 not included earlier in the sequence.
%Y A096098 Cf. A096097.
%K A096098 base,nonn
%O A096098 1,1
%A A096098 _Amarnath Murthy_, Jun 24 2004
%E A096098 More terms from _R. J. Mathar_, Aug 03 2007
%E A096098 a(23)-a(26) from _N. J. A. Sloane_, Nov 10 2007
%E A096098 Corrected and extended by _Martin Fuller_, Nov 21 2007
%E A096098 More terms from _Sean A. Irvine_, May 25 2010
%E A096098 Example detailed. - _Wolfdieter Lang_, May 08 2014